BunchNielsenSorensen formulaFrom Wikipedia, the free encyclopedia

Contents

1 BackusGilbert method 11.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Balanced set 22.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Barycentric coordinate system 43.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 Barycentric coordinates on triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3.2.1 Conversion between barycentric and Cartesian coordinates . . . . . . . . . . . . . . . . . 63.2.2 Conversion between barycentric and trilinear coordinates . . . . . . . . . . . . . . . . . . 73.2.3 Application: Determining location with respect to a triangle . . . . . . . . . . . . . . . . . 73.2.4 Application: Interpolation on a triangular unstructured grid . . . . . . . . . . . . . . . . . 73.2.5 Application: Integration over a triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.3 Barycentric coordinates on tetrahedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.4 Generalized barycentric coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.4.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Basis (linear algebra) 114.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2 Expression of a basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.5 Extending to a basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.6 Example of alternative proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.6.1 From the denition of basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

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4.6.2 By the dimension theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.6.3 By the invertible matrix theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.7 Ordered bases and coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.8 Related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.8.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.8.2 Ane geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.9 Proof that every vector space has a basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.10 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.12.1 General references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.12.2 Historical references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.13 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5 Basis function 225.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.1.1 Polynomial bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.1.2 Fourier basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

6 Bidiagonal matrix 236.1 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

7 Big M method 257.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257.2 Other usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.4 References and external links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

8 Bilinear form 278.1 Coordinate representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278.2 Maps to the dual space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278.3 Symmetric, skew-symmetric and alternating forms . . . . . . . . . . . . . . . . . . . . . . . . . . 288.4 Derived quadratic form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.5 Reexivity and orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.6 Dierent spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.7 Relation to tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308.8 On normed vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

CONTENTS iii

8.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

9 Binomial inverse theorem 339.1 Verication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339.2 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

9.2.1 First . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349.2.2 Second . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349.2.3 Third . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

9.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

10 Braket notation 3610.1 Vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

10.1.1 Background: Vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.1.2 Ket notation for vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.1.3 Inner products and bras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3810.1.4 Non-normalizable states and non-Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . 39

10.2 Usage in quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4010.2.1 Spinless positionspace wave function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4010.2.2 Overlap of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4110.2.3 Changing basis for a spin-1/2 particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4210.2.4 Misleading uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

10.3 Linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4310.3.1 Linear operators acting on kets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4310.3.2 Linear operators acting on bras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4310.3.3 Outer products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4310.3.4 Hermitian conjugate operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

10.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4410.4.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4410.4.2 Associativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4510.4.3 Hermitian conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

10.5 Composite bras and kets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.6 The unit operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.7 Notation used by mathematicians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.9 References and notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4810.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

11 BunchNielsenSorensen formula 49

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11.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4911.2 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4911.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4911.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4911.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4911.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5011.7 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 51

11.7.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5111.7.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5211.7.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Chapter 1

BackusGilbert method

In mathematics, the BackusGilbert method, also known as the optimally localized average (OLA) method isnamed for its discoverers, geophysicists George E. Backus and James Freeman Gilbert. It is a regularization methodfor obtaining meaningful solutions to ill-posed inverse problems. Where other regularization methods, such as thefrequently used Tikhonov regularization method, seek to impose smoothness constraints on the solution, BackusGilbert instead seeks to impose stability constraints, so that the solution would vary as little as possible if the inputdata were resampled multiple times. In practice, and to the extent that is justied by the data, smoothness resultsfrom this.

1.1 References Backus, G.E., and Gilbert, F. 1968, The Resolving power of Gross Earth Data, Geophysical Journal of theRoyal Astronomical Society, vol. 16, pp. 169205.

Backus, G.E., and Gilbert, F. 1970, Uniqueness in the Inversion of inaccurate Gross Earth Data, Philosoph-ical Transactions of the Royal Society of London A, vol. 266, pp. 123192.

Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). Section 19.6. BackusGilbert Method.Numerical Recipes: The Art of Scientic Computing (3rd ed.). New York: Cambridge University Press. ISBN978-0-521-88068-8.

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Chapter 2

Balanced set

In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a eldK with an absolute value function | |) is a set S such that for all scalars with || 1

S S

where

S := fx j x 2 Sg:

The balanced hull or balanced envelope for a set S is the smallest balanced set containing S. It can be constructedas the intersection of all balanced sets containing S.

2.1 Examples The open and closed balls in a normed vector space are balanced sets.

Any subspace of a real or complex vector space is a balanced set.

The cartesian product of a family of balanced sets is balanced in the product space of the corresponding vectorspaces (over the same eld K).

Consider , the eld of complex numbers, as a 1-dimensional vector space. The balanced sets are itself, theempty set and the open and closed discs centered at 0 (visualizing complex numbers as points in the plane).Contrariwise, in the two dimensional Euclidean space there are many more balanced sets: any line segmentwith midpoint at (0,0) will do. As a result, and 2 are entirely dierent as far as their vector space structureis concerned.

If p is a semi-norm on a linear space X, then for any constant c>0, the set {x X | p(x)c} is balanced.

2.2 Properties The union and intersection of balanced sets is a balanced set.

The closure of a balanced set is balanced.

By denition (not property), a set is absolutely convex if and only if it is convex and balanced.

Every balanced set is a symmetric set

2

2.3. SEE ALSO 3

2.3 See also Star domain

2.4 References Robertson, A.P.; W.J. Robertson (1964). Topological vector spaces. Cambridge Tracts in Mathematics 53.Cambridge University Press. p. 4.

W. Rudin (1990). Functional Analysis (2nd ed ed.). McGraw-Hill, Inc. ISBN 0-07-054236-8. H.H. Schaefer (1970). Topological Vector Spaces. GTM 3. Springer-Verlag. p. 11. ISBN 0-387-05380-8.

Chapter 3

Barycentric coordinate system

Not to be confused with Barycentric coordinates (astronomy).

In geometry, the barycentric coordinate system is a coordinate system in which the location of a point of a simplex(a triangle, tetrahedron, etc.) is specied as the center of mass, or barycenter, of masses placed at its vertices.Coordinates also extend outside the simplex, where one or more coordinates become negative. The system wasintroduced (1827) by August Ferdinand Mbius.

3.1 DenitionLet x1; : : : ; xn be the vertices of a simplex in an ane space A. If, for some point p in A,

(a1 + + an)p = a1 x1 + + an xnand at least one of a1; : : : ; an does not vanish then we say that the coecients ( a1; : : : ; an ) are barycentric coor-dinates of p with respect to x1; : : : ; xn . The vertices themselves have the coordinates x1 = (1; 0; 0; : : : ; 0); x2 =(0; 1; 0; : : : ; 0); : : : ; xn = (0; 0; 0; : : : ; 1) . Barycentric coordinates are not unique: for any b not equal to zero, (ba1; : : : ; ban ) are also barycentric coordinates of p.When the coordinates are not negative, the point p lies in the convex hull of x1; : : : ; xn , that is, in the simplex whichhas those points as its vertices.Barycentric coordinates, as dened above, are a form of homogeneous coordinates. Sometimes values of coordinatesare restricted with a condition

Xai = 1

which makes them unique; then, they are ane coordinates.

3.2 Barycentric coordinates on trianglesSee also: Ternary plot and Triangle centerIn the context of a triangle, barycentric coordinates are also known as area coordinates or areal coordinates,because the coordinates of P with respect to triangle ABC are equivalent to the (signed) ratios of the areas of PBC,PCA and PAB to the area of the reference triangle ABC. Areal and trilinear coordinates are used for similar purposesin geometry.Barycentric or areal coordinates are extremely useful in engineering applications involving triangular subdomains.These make analytic integrals often easier to evaluate, and Gaussian quadrature tables are often presented in termsof area coordinates.

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3.2. BARYCENTRIC COORDINATES ON TRIANGLES 5

(1,0,0) (0,1,0)

(0,0,1)

(1/2,1/2,0)

(1/2,0,1/2) (0,1/2,1/2)(1/4,1/4,1/2)

(1/4,1/2,1/4)(1/2,1/4,1/4)

(1/3,1/3,1/3)

(1,0,0) (0,1,0)

(0,0,1)

(1/2,1/2,0)

(1/2,0,1/2) (0,1/2,1/2)(1/4,1/4,1/2)

(1/4,1/2,1/4)(1/2,1/4,1/4)

(1/3,1/3,1/3)

Barycentric coordinates (1; 2; 3) on an equilateral triangle and on a right triangle.

6 CHAPTER 3. BARYCENTRIC COORDINATE SYSTEM

Consider a triangle T dened by its three vertices, r1 , r2 and r3 . Each point r located inside this triangle can bewritten as a unique convex combination of the three vertices. In other words, for each r there is a unique sequenceof three numbers, 1; 2; 3 0 such that 1 + 2 + 3 = 1 and

r = 1r1 + 2r2 + 3r3;

The three numbers 1; 2; 3 indicate the barycentric or area coordinates of the point r with respect to thetriangle. They are often denoted as ; ; instead of 1; 2; 3 . Note that although there are three coordinates,there are only two degrees of freedom, since 1 + 2 + 3 = 1 . Thus every point is uniquely dened by any two ofthe barycentric coordinates.Switching back and forth between the barycentric coordinates and other coordinate systems makes some problemsmuch easier to solve.

3.2.1 Conversion between barycentric and Cartesian coordinatesGiven a point r inside a triangle one can obtain the barycentric coordinates 1 , 2 and 3 from the Cartesiancoordinates (x; y) or vice versa.We can write the Cartesian coordinates of the point r in terms of the Cartesian components of the triangle verticesr1 , r2 , r3 where ri = (xi; yi) and in terms of the barycentric coordinates of r as

x = 1x1 + 2x2 + 3x3y = 1y1 + 2y2 + 3y3

To nd the reverse transformation, from Cartesian coordinates to barycentric coordinates, we rst substitute 3 =1 1 2 into the above to obtain

x = 1x1 + 2x2 + (1 1 2)x3y = 1y1 + 2y2 + (1 1 2)y3Rearranging, this is

1(x1 x3) + 2(x2 x3) + x3 x = 01(y1 y3) + 2(y2 y3) + y3 y = 0This linear transformation may be written more succinctly as

T = r r3where is the vector of barycentric coordinates, r is the vector of Cartesian coordinates, and T is a matrix given by

T =x1 x3 x2 x3y1 y3 y2 y3

Now the matrix T is invertible, since r1 r3 and r2 r3 are linearly independent (if this were not the case, then r1, r2 , and r3 would be collinear and would not form a triangle). Thus, we can rearrange the above equation to get

12

= T1(r r3)

Finding the barycentric coordinates has thus been reduced to nding the inverse matrix of T , an easy problem in thecase of 22 matrices.

3.2. BARYCENTRIC COORDINATES ON TRIANGLES 7

Explicitly, the formulae for the barycentric co-ordinates of point r in terms of its Cartesian coordinates (x, y) and interms of the Cartesian coordinates of the triangles vertices are:

1 =(y2 y3)(x x3) + (x3 x2)(y y3)

det(T ) =(y2 y3)(x x3) + (x3 x2)(y y3)(y2 y3)(x1 x3) + (x3 x2)(y1 y3) ;

2 =(y3 y1)(x x3) + (x1 x3)(y y3)

det(T ) =(y3 y1)(x x3) + (x1 x3)(y y3)(y2 y3)(x1 x3) + (x3 x2)(y1 y3) ;

3 = 1 1 2 :

3.2.2 Conversion between barycentric and trilinear coordinatesA point with trilinear coordinates x : y : z has barycentric coordinates ax : by : cz where a, b, c are the sidelengths ofthe triangle. Conversely, a point with barycentrics : : has trilinears /a : /b : /c.

3.2.3 Application: Determining location with respect to a triangleAlthough barycentric coordinates are most commonly used to handle points inside a triangle, they can also be used todescribe a point outside the triangle. If the point is not inside the triangle, then we can still use the formulas above tocompute the barycentric coordinates. However, since the point is outside the triangle, at least one of the coordinateswill violate our original assumption that 1:::3 0 . In fact, given any point in cartesian coordinates, we can use thisfact to determine where this point is with respect to a triangle.If a point lies in the interior of the triangle, all of the Barycentric coordinates lie in the open interval (0; 1): If apoint lies on an edge of the triangle but not at a vertex, one of the area coordinates 1:::3 (the one associated with theopposite vertex) is zero, while the other two lie in the open interval (0; 1): If the point lies on a vertex, the coordinateassociated with that vertex equals 1 and the others equal zero. Finally, if the point lies outside the triangle at leastone coordinate is negative.Summarizing,

Point r lies inside the triangle if and only if 0 < i < 1 8 i in 1; 2; 3 .Otherwise, r lies on the edge or corner of the triangle if 0 i 1 8 i in 1; 2; 3 .Otherwise, r lies outside the triangle.

In particular, if a point lies on the opposite side of a sideline from the vertex opposite that sideline, then that pointsbarycentric coordinate corresponding to that vertex is negative.

3.2.4 Application: Interpolation on a triangular unstructured gridIf f(r1); f(r2); f(r3) are known quantities, but the values of f inside the triangle dened by r1; r2; r3 is unknown,we can approximate these values using linear interpolation. Barycentric coordinates provide a convenient way tocompute this interpolation. If r is a point inside the triangle with barycentric coordinates 1 , 2 , 3 , then

f(r) 1f(r1) + 2f(r2) + 3f(r3)

In general, given any unstructured grid or polygon mesh, we can use this kind of technique to approximate the valueof f at all points, as long as the functions value is known at all vertices of the mesh. In this case, we have manytriangles, each corresponding to a dierent part of the space. To interpolate a function f at a point r , we must rstnd a triangle that contains it. To do so, we rst transform r into the barycentric coordinates of each triangle. If wend some triangle such that the coordinates satisfy 0 i 1 8 i in 1; 2; 3 , then the point lies in that triangle or onits edge (explained in the previous section). We can then interpolate the value of f(r) as described above.These methods have many applications, such as the nite element method (FEM).

8 CHAPTER 3. BARYCENTRIC COORDINATE SYSTEM

3.2.5 Application: Integration over a triangleThe integral of a function over the domain of the triangle can be annoying to compute in a cartesian coordinatesystem. One generally has to split the triangle up into two halves, and great messiness follows. Instead, it is ofteneasier to make a change of variables to any two barycentric coordinates, e.g. 1; 2 . Under this change of variables,

ZT

f(r) dr = 2AZ 10

Z 120

f(1r1 + 2r2 + (1 1 2)r3) d1 d2

where A is the area of the triangle. This result follows from the fact that a rectangle in barycentric coordinatescorresponds to a quadrilateral in cartesian coordinates, and the ratio of the areas of the corresponding shapes in thecorresponding coordinate systems is given by 2A .

3.2.6 ExamplesThe circumcenter of a triangle ABC has barycentric coordinates[1][2]

a2(a2 + b2 + c2) : b2(a2 b2 + c2) : c2(a2 + b2 c2)= sin 2A : sin 2B : sin 2C;where a, b, c are edge lengths BC, CA, AB respectively of the triangle.The orthocenter has barycentric coordinates

tanA : tanB : tanC:The incenter has barycentric coordinates

a : b : c = sinA : sinB : sinC:The nine-point center has barycentric coordinates

a cos(B C) : b cos(C A) : c cos(AB)= a2(b2 + c2) (b2 c2)2 : b2(c2 + a2) (c2 a2)2 : c2(a2 + b2) (a2 b2)2:

3.3 Barycentric coordinates on tetrahedraBarycentric coordinates may be easily extended to three dimensions. The 3D simplex is a tetrahedron, a polyhedronhaving four triangular faces and four vertices. Once again, the barycentric coordinates are dened so that the rstvertex r1 maps to barycentric coordinates = (1; 0; 0; 0) , r2 ! (0; 1; 0; 0) , etc.This is again a linear transformation, and we may extend the above procedure for triangles to nd the barycentriccoordinates of a point r with respect to a tetrahedron:

0@123

1A = T1(r r4)where T is now a 33 matrix:

T =

0@x1 x4 x2 x4 x3 x4y1 y4 y2 y4 y3 y4z1 z4 z2 z4 z3 z4

1A

3.4. GENERALIZED BARYCENTRIC COORDINATES 9

Once again, the problem of nding the barycentric coordinates has been reduced to inverting a 33 matrix. 3Dbarycentric coordinates may be used to decide if a point lies inside a tetrahedral volume, and to interpolate a functionwithin a tetrahedral mesh, in an analogous manner to the 2D procedure. Tetrahedral meshes are often used in niteelement analysis because the use of barycentric coordinates can greatly simplify 3D interpolation.

3.4 Generalized barycentric coordinatesBarycentric coordinates (a1, ..., an) that are dened with respect to a polytope instead of a simplex are called gener-alized barycentric coordinates. For these, the equation

(a1 + + an)p = a1x1 + + anxnis still required to hold where x1, ..., xn are the vertices of the given polytope. Thus, the denition is formallyunchanged but while a simplex with n vertices needs to be embedded in a vector space of dimension of at least n-1,a polytope may be embedded in a vector space of lower dimension. The simplest example is a quadrilateral in theplane. Consequently, even normalized generalized barycentric coordinates (i.e. coordinates such that the sum of thecoecients is 1) are in general not uniquely determined anymore while this is the case for normalized barycentriccoordinates with respect to a simplex.More abstractly, generalized barycentric coordinates express a polytope with n vertices, regardless of dimension, asthe image of the standard (n 1) -simplex, which has n vertices the map is onto: n1 P: The map is one-to-one if and only if the polytope is a simplex, in which case the map is an isomorphism; this corresponds to a point nothaving unique generalized barycentric coordinates except when P is a simplex.Dual to generalized barycentric coordinates are slack variables, which measure by how much margin a point satisesthe linear constraints, and gives an embedding P ,! (R0)f into the f-orthant, where f is the number of faces (dualto the vertices). This map is one-to-one (slack variables are uniquely determined) but not onto (not all combinationscan be realized).This use of the standard (n 1) -simplex and f-orthant as standard objects that map to a polytope or that a polytopemaps into should be contrasted with the use of the standard vector spaceKn as the standard object for vector spaces,and the standard ane hyperplane f(x0; : : : ; xn) j

Pxi = 1g Kn+1 as the standard object for ane spaces,

where in each case choosing a linear basis or ane basis provides an isomorphism, allowing all vector spaces andane spaces to be thought of in terms of these standard spaces, rather than an onto or one-to-one map (not everypolytope is a simplex). Further, the n-orthant is the standard object that maps to cones.

3.4.1 Applications

Generalized barycentric coordinates have applications in computer graphics and more specically in geometric mod-elling. Often, a three-dimensional model can be approximated by a polyhedron such that the generalized barycentriccoordinates with respect to that polyhedron have a geometric meaning. In this way, the processing of the model canbe simplied by using these meaningful coordinates.

3.5 See also Ternary plot Convex combination

3.6 References[1] Wolfram page on barycentric coordinates

[2] Clark Kimberlings Encyclopedia of Triangles http://faculty.evansville.edu/ck6/encyclopedia/ETC.html

10 CHAPTER 3. BARYCENTRIC COORDINATE SYSTEM

Bradley, Christopher J. (2007). The Algebra of Geometry: Cartesian, Areal and Projective Co-ordinates. Bath:Highperception. ISBN 978-1-906338-00-8.

Coxeter, H.S.M. (1969). Introduction to geometry (2nd ed.). John Wiley and Sons. pp. 216221. ISBN978-0-471-50458-0. Zbl 0181.48101.

Barycentric Calculus In Euclidean And Hyperbolic Geometry: A Comparative Introduction, Abraham Ungar,World Scientic, 2010

Hyperbolic Barycentric Coordinates, Abraham A. Ungar, The Australian Journal of Mathematical Analysisand Applications, Vol.6, No.1, Article 18, pp. 1-35, 2009

Weisstein, Eric W., Areal Coordinates, MathWorld. Weisstein, Eric W., Barycentric Coordinates, MathWorld.

3.7 External links The uses of homogeneous barycentric coordinates in plane euclidean geometry Barycentric Coordinates a collection of scientic papers about (generalized) barycentric coordinates Barycentric coordinates: A Curious Application (solving the three glasses problem) at cut-the-knot Accurate point in triangle test Barycentric Coordinates in Olympiad Geometry by Evan Chen and Max Schindler

Chapter 4

Basis (linear algebra)

Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a moregeneral concept in physics, see frame of reference.

A set of vectors in a vector space V is called a basis, or a set of basis vectors, if the vectors are linearly independentand every other vector in the vector space is linearly dependent on these vectors.[1] In more general terms, a basis isa linearly independent spanning set.Given a basis of a vector space V, every element of V can be expressed uniquely as a linear combination of basisvectors, whose coecients are referred to as vector coordinates or components. A vector space can have severaldistinct sets of basis vectors; however each such set has the same number of elements, with this number being thedimension of the vector space.

4.1 DenitionA basis B of a vector space V over a eld F is a linearly independent subset of V that spans V.In more detail, suppose that B = { v1, , vn } is a nite subset of a vector space V over a eld F (such as the real orcomplex numbers R or C). Then B is a basis if it satises the following conditions:

the linear independence property,

for all a1, , an F, if a1v1 + + anvn = 0, then necessarily a1 = = an = 0; and

the spanning property,

for every x in V it is possible to choose a1, , an F such that x = a1v1 + + anvn.

The numbers a are called the coordinates of the vector x with respect to the basis B, and by the rst property theyare uniquely determined.A vector space that has a nite basis is called nite-dimensional. To deal with innite-dimensional spaces, we mustgeneralize the above denition to include innite basis sets. We therefore say that a set (nite or innite) B V is abasis, if

every nite subset B0 B obeys the independence property shown above; and for every x in V it is possible to choose a1, , an F and v1, , vn B such that x = a1v1 + + anvn.

The sums in the above denition are all nite because without additional structure the axioms of a vector space donot permit us to meaningfully speak about an innite sum of vectors. Settings that permit innite linear combinationsallow alternative denitions of the basis concept: see Related notions below.It is often convenient to list the basis vectors in a specic order, for example, when considering the transformationmatrix of a linear map with respect to a basis. We then speak of an ordered basis, which we dene to be a sequence(rather than a set) of linearly independent vectors that span V: see Ordered bases and coordinates below.

11

12 CHAPTER 4. BASIS (LINEAR ALGEBRA)

4.2 Expression of a basisThere are several ways to describe a basis for the space. Some are made ad hoc for a specic dimension. For example,there are several ways to give a basis in dim 3, like Euler angles.The general case is to give a matrix with the components of the new basis vectors in columns. This is also the moregeneral method because it can express any possible set of vectors even if it is not a basis. This matrix can be seen asthree things:Basis matrix: Is a matrix that represents the basis, because its columns are the components of vectors of the basis.This matrix represents any vector of the new basis as linear combination of the current basis.Rotation operator: When orthonormal bases are used, any other orthonormal basis can be dened by a rotationmatrix. This matrix represents the rotation operator that rotates the vectors of the basis to the new one. It is exactlythe same matrix as before because the rotation matrix multiplied by the identity matrix I has to be the new basismatrix.Change of basis matrix: This matrix can be used to change dierent objects of the space to the new basis. Thereforeis called "change of basis" matrix. It is important to note that some objects change their components with this matrixand some others, like vectors, with its inverse.

4.3 PropertiesAgain, B denotes a subset of a vector space V. Then, B is a basis if and only if any of the following equivalentconditions are met:

B is a minimal generating set of V, i.e., it is a generating set and no proper subset of B is also a generating set.

B is a maximal set of linearly independent vectors, i.e., it is a linearly independent set but no other linearlyindependent set contains it as a proper subset.

Every vector in V can be expressed as a linear combination of vectors in B in a unique way. If the basisis ordered (see Ordered bases and coordinates below) then the coecients in this linear combination providecoordinates of the vector relative to the basis.

Every vector space has a basis. The proof of this requires the axiom of choice. All bases of a vector space havethe same cardinality (number of elements), called the dimension of the vector space. This result is known as thedimension theorem, and requires the ultralter lemma, a strictly weaker form of the axiom of choice.Also many vector sets can be attributed a standard basis which comprises both spanning and linearly independentvectors.Standard bases for example:In Rn {E1,...,En} where En is the n-th column of the identity matrix which consists of all ones in the main diagonaland zeros everywhere else. This is because the columns of the identity matrix are linearly independent can alwaysspan a vector set by expressing it as a linear combination.In P2 where P2 is the set of all polynomials of degree at most 2 {1,x,x2} is the standard basis.In M22 {M,,M,,M,,M,} where M22 is the set of all 22 matrices. and M, is the 22 matrix with a 1 in them,n position and zeros everywhere else. This again is a standard basis since it is linearly independent and spanning.

4.4 Examples Consider R2, the vector space of all coordinates (a, b) where both a and b are real numbers. Then a verynatural and simple basis is simply the vectors e1 = (1,0) and e2 = (0,1): suppose that v = (a, b) is a vector inR2, then v = a (1,0) + b (0,1). But any two linearly independent vectors, like (1,1) and (1,2), will also forma basis of R2.

4.5. EXTENDING TO A BASIS 13

More generally, the vectors e1, e2, ..., en are linearly independent and generate Rn. Therefore, they form abasis for Rn and the dimension of Rn is n. This basis is called the standard basis.

Let V be the real vector space generated by the functions et and e2t . These two functions are linearly indepen-dent, so they form a basis for V.

Let R[x] denote the vector space of real polynomials; then (1, x, x2, ...) is a basis of R[x]. The dimension ofR[x] is therefore equal to aleph-0.

4.5 Extending to a basisLet S be a subset of a vector space V. To extend S to a basis means to nd a basis B that contains S as a subset. Thiscan be done if and only if S is linearly independent. Almost always, there is more than one such B, except in ratherspecial circumstances (i.e. S is already a basis, or S is empty and V has two elements).A similar question is when does a subset S contain a basis. This occurs if and only if S spans V. In this case, S willusually contain several dierent bases.

4.6 Example of alternative proofsOften, a mathematical result can be proven in more than one way. Here, using three dierent proofs, we show thatthe vectors (1,1) and (1,2) form a basis for R2.

4.6.1 From the denition of basis

We have to prove that these two vectors are linearly independent and that they generate R2.Part I: If two vectors v,w are linearly independent, then av + bw = 0 (a and b scalars) implies a = 0; b = 0:To prove that they are linearly independent, suppose that there are numbers a,b such that:

a(1; 1) + b(1; 2) = (0; 0)

(i.e., they are linearly dependent). Then:

(a b; a+ 2b) = (0; 0)anda b = 0anda+ 2b = 0:

Subtracting the rst equation from the second, we obtain:

3b = 0sob = 0:

Adding this equation to the rst equation then:

14 CHAPTER 4. BASIS (LINEAR ALGEBRA)

a = 0:

Hence we have linear independence.Part II: To prove that these two vectors generate R2, we have to let (a,b) be an arbitrary element of R2, and showthat there exist numbers r,s R such that:

r(1; 1) + s(1; 2) = (a; b):Then we have to solve the equations:

r s = ar + 2s = b:

Subtracting the rst equation from the second, we get:

3s = b a;and thens = (b a)/3;and nallyr = s+ a = ((b a)/3) + a = (b+ 2a)/3:

4.6.2 By the dimension theoremSince (1,2) is clearly not a multiple of (1,1) and since (1,1) is not the zero vector, these two vectors are linearlyindependent. Since the dimension of R2 is 2, the two vectors already form a basis of R2 without needing anyextension.

4.6.3 By the invertible matrix theoremSimply compute the determinant

det1 11 2

= 3 6= 0:

Since the above matrix has a nonzero determinant, its columns form a basis of R2. See: invertible matrix.

4.7 Ordered bases and coordinatesA basis is just a linearly independent set of vectors with or without a given ordering. For many purposes it is conve-nient to work with an ordered basis. For example, when working with a coordinate representation of a vector it iscustomary to speak of the rst or second coordinate, which makes sense only if an ordering is specied for thebasis. For nite-dimensional vector spaces one typically indexes a basis {vi} by the rst n integers. An ordered basisis also called a frame.Suppose V is an n-dimensional vector space over a eld F. A choice of an ordered basis for V is equivalent to a choiceof a linear isomorphism from the coordinate space Fn to V.Proof. The proof makes use of the fact that the standard basis of Fn is an ordered basis.Suppose rst that

4.8. RELATED NOTIONS 15

: Fn V

is a linear isomorphism. Dene an ordered basis {vi} for V by

vi = (ei) for 1 i n

where {ei} is the standard basis for Fn.Conversely, given an ordered basis, consider the map dened by

(x) = x1v1 + x2v2 + ... + xnvn,

where x = x1e1 + x2e2 + ... + xnen is an element of Fn. It is not hard to check that is a linear isomorphism.These two constructions are clearly inverse to each other. Thus ordered bases for V are in 1-1 correspondence withlinear isomorphisms Fn V.The inverse of the linear isomorphism determined by an ordered basis {vi} equips V with coordinates: if, for avector v V, 1(v) = (a1, a2,...,an) Fn, then the components aj = aj(v) are the coordinates of v in the sense thatv = a1(v) v1 + a2(v) v2 + ... + an(v) vn.The maps sending a vector v to the components aj(v) are linear maps from V to F, because of 1 is linear. Hencethey are linear functionals. They form a basis for the dual space of V, called the dual basis.

4.8 Related notions

4.8.1 AnalysisIn the context of innite-dimensional vector spaces over the real or complex numbers, the term Hamel basis (namedafter Georg Hamel) or algebraic basis can be used to refer to a basis as dened in this article. This is to make adistinction with other notions of basis that exist when innite-dimensional vector spaces are endowed with extrastructure. The most important alternatives are orthogonal bases on Hilbert spaces, Schauder bases and Markushevichbases on normed linear spaces. The term Hamel basis is also commonly used to mean a basis for the real numbersR as a vector space over the eld Q of rational numbers. (In this case, the dimension of R over Q is uncountable,specically the continuum, the cardinal number 2.)The common feature of the other notions is that they permit the taking of innite linear combinations of the basicvectors in order to generate the space. This, of course, requires that innite sums are meaningfully dened on thesespaces, as is the case for topological vector spaces a large class of vector spaces including e.g. Hilbert spaces,Banach spaces or Frchet spaces.The preference of other types of bases for innite-dimensional spaces is justied by the fact that the Hamel basisbecomes too big in Banach spaces: If X is an innite-dimensional normed vector space which is complete (i.e. X isa Banach space), then any Hamel basis of X is necessarily uncountable. This is a consequence of the Baire categorytheorem. The completeness as well as innite dimension are crucial assumptions in the previous claim. Indeed, nite-dimensional spaces have by denition nite bases and there are innite-dimensional (non-complete) normed spaceswhich have countable Hamel bases. Consider c00 , the space of the sequences x = (xn) of real numbers which haveonly nitely many non-zero elements, with the norm kxk = supn jxnj: Its standard basis, consisting of the sequenceshaving only one non-zero element, which is equal to 1, is a countable Hamel basis.

Example

In the study of Fourier series, one learns that the functions {1} { sin(nx), cos(nx) : n = 1, 2, 3, ... } are an orthogonalbasis of the (real or complex) vector space of all (real or complex valued) functions on the interval [0, 2] that aresquare-integrable on this interval, i.e., functions f satisfying

Z 20

jf(x)j2 dx

16 CHAPTER 4. BASIS (LINEAR ALGEBRA)

The functions {1} { sin(nx), cos(nx) : n = 1, 2, 3, ... } are linearly independent, and every function f that issquare-integrable on [0, 2] is an innite linear combination of them, in the sense that

limn!1

Z 20

a0 + nXk=1

ak cos(kx) + bk sin(kx)

f(x)2 dx = 0for suitable (real or complex) coecients ak, bk. But most square-integrable functions cannot be represented as nitelinear combinations of these basis functions, which therefore do not comprise a Hamel basis. Every Hamel basis ofthis space is much bigger than this merely countably innite set of functions. Hamel bases of spaces of this kind aretypically not useful, whereas orthonormal bases of these spaces are essential in Fourier analysis.

4.8.2 Ane geometry

The related notions of an ane space, projective space, convex set, and cone have related notions of ane basis[2](a basis for an n-dimensional ane space is n+ 1 points in general linear position), projective basis (essentially thesame as an ane basis, this is n + 1 points in general linear position, here in projective space), convex basis (thevertices of a polytope), and cone basis[3] (points on the edges of a polygonal cone); see also a Hilbert basis (linearprogramming).

4.9 Proof that every vector space has a basisLet V be any vector space over some eld F. Every vector space must contain at least one element: the zero vector 0.Note that if V = {0}, then the empty set is a basis for V. Now we consider the case where V contains at least onenonzero element, say v.Dene the setX as all linear independent subsets ofV. Note that sinceV contains the nonzero element v, the singletonsubset L = {v} of V is necessarily linearly independent.Hence the set X contains at least the subset L = {v}, and so X is nonempty.We let X be partially ordered by inclusion: If L1 and L2 belong to X, we say that L1 L2 when L1 L2. It is easyto check that (X, ) satises the denition of a partially ordered set.We now note that if Y is a subset of X that is totally ordered by , then the union LY of all the elements of Y (whichare themselves certain subsets of V) is an upper bound for Y. To show this, it is necessary to verify both that a) LYbelongs to X, and that b) every element L of Y satises L LY. Both a) and b) are easy to check.Now we apply Zorns lemma, which asserts that because X is nonempty, and every totally ordered subset of thepartially ordered set (X, ) has an upper bound, it follows thatX has a maximal element. (In other words, there existssome element L of X satisfying the condition that whenever L L for some element L of X, then L = L.)Finally we claim that L is a basis forV. Since L belongs toX, we already know that L is a linearly independentsubset of V.Now suppose L does not span V. Then there exists some vector w of V that cannot be expressed as a linearlycombination of elements of L (with coecients in the eld F). Note that such a vector w cannot be an element ofL.Now consider the subset L of V dened by L = L {w}. It is easy to see that a) L L (since L is asubset of L), and that b) L L (because L contains the vector w that is not contained in L).But the combination of a) and b) above contradict the fact that L is a maximal element of X, which we havealready proved. This contradiction shows that the assumption that L does not span V was not true.Hence L does span V. Since we also know that L is linearly independent over the eld F, this veries that Lis a basis for V. Which proves that the arbitrary vector space V has a basis.Note: This proof relies on Zorns lemma, which is logically equivalent to the Axiom of Choice. It turns out that,conversely, the assumption that every vector space has a basis can be used to prove the Axiom of Choice. Thus thetwo assertions are logically equivalent.

4.10. SEE ALSO 17

4.10 See also Change of basis Frame of a vector space Spherical basis

4.11 Notes[1] Halmos, Paul Richard (1987) Finite-dimensional vector spaces (4th edition) Springer-Verlag, New York, page 10, ISBN

0-387-90093-4

[2] Notes on geometry, by Elmer G. Rees, p. 7

[3] Some remarks about additive functions on cones, Marek Kuczma

4.12 References

4.12.1 General references Blass, Andreas (1984), Existence of bases implies the axiom of choice, Axiomatic set theory, ContemporaryMathematics volume 31, Providence, R.I.: AmericanMathematical Society, pp. 3133, ISBN 0-8218-5026-1,MR 763890

Brown, William A. (1991), Matrices and vector spaces, New York: M. Dekker, ISBN 978-0-8247-8419-5 Lang, Serge (1987), Linear algebra, Berlin, New York: Springer-Verlag, ISBN 978-0-387-96412-6

4.12.2 Historical references Banach, Stefan (1922), Sur les oprations dans les ensembles abstraits et leur application aux quations int-grales (On operations in abstract sets and their application to integral equations)" (PDF), Fundamenta Mathe-maticae (in French) 3, ISSN 0016-2736

Bolzano, Bernard (1804), Betrachtungen ber einige Gegenstnde der Elementargeometrie (Considerations ofsome aspects of elementary geometry) (in German)

Bourbaki, Nicolas (1969), lments d'histoire des mathmatiques (Elements of history of mathematics) (inFrench), Paris: Hermann

Dorier, Jean-Luc (1995), A general outline of the genesis of vector space theory, Historia Mathematica 22(3): 227261, doi:10.1006/hmat.1995.1024, MR 1347828

Fourier, Jean Baptiste Joseph (1822), Thorie analytique de la chaleur (in French), Chez Firmin Didot, preet ls

Grassmann, Hermann (1844), Die Lineale Ausdehnungslehre - Ein neuer Zweig der Mathematik (in German),reprint: Hermann Grassmann. Translated by Lloyd C. Kannenberg. (2000), Extension Theory, Kannenberg,L.C., Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2031-5

Hamilton, William Rowan (1853), Lectures on Quaternions, Royal Irish Academy Mbius, August Ferdinand (1827), Der Barycentrische Calcul : ein neues Hlfsmittel zur analytischen Behand-lung der Geometrie (Barycentric calculus: a new utility for an analytic treatment of geometry) (in German)

Moore, Gregory H. (1995), The axiomatization of linear algebra: 18751940, Historia Mathematica 22 (3):262303, doi:10.1006/hmat.1995.1025

Peano, Giuseppe (1888), Calcolo Geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalleOperazioni della Logica Deduttiva (in Italian), Turin

18 CHAPTER 4. BASIS (LINEAR ALGEBRA)

4.13 External links Instructional videos from Khan Academy

Introduction to bases of subspaces Proof that any subspace basis has same number of elements

Hazewinkel, Michiel, ed. (2001), Basis, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

4.13. EXTERNAL LINKS 19

The same vector can be represented in two dierent bases (purple and red arrows).

20 CHAPTER 4. BASIS (LINEAR ALGEBRA)

1 2012

1

1

(2,1)(0,1)

(1,0)x

y

This picture illustrates the standard basis in R2. The blue and orange vectors are the elements of the basis; the green vector can begiven in terms of the basis vectors, and so is linearly dependent upon them.

4.13. EXTERNAL LINKS 21

N

x

y

z

Z

X

Y

Basis dened by Euler angles - The xyz (xed) system is shown in blue, the XYZ (rotated) system is shown in red. The line of nodes,labeled N, is shown in green.

Chapter 5

Basis function

In mathematics, a basis function is an element of a particular basis for a function space. Every continuous functionin the function space can be represented as a linear combination of basis functions, just as every vector in a vectorspace can be represented as a linear combination of basis vectors.In numerical analysis and approximation theory, basis functions are also called blending functions, because of theiruse in interpolation: In this application, a mixture of the basis functions provides an interpolating function (with theblend depending on the evaluation of the basis functions at the data points).

5.1 Examples

5.1.1 Polynomial basesThe collection of quadratic polynomials with real coecients has {1, t, t2} as a basis. Every quadratic polynomialcan be written as a1+bt+ct2, that is, as a linear combination of the basis functions 1, t, and t2. The set {(t1)(t2)/2,t(t2), t(t1)/2} is another basis for quadratic polynomials, called the Lagrange basis. The rst three Chebyshevpolynomials form yet another basis

5.1.2 Fourier basisSines and cosines form an (orthonormal) Schauder basis for square-integrable functions. As a particular example, thecollection:

fp2 sin(2nx) j n 2 Ng [ f

p2 cos(2nx) j n 2 Ng [ f1g

forms a basis for L2(0,1).

5.2 References Ito, Kiyoshi (1993). Encyclopedic Dictionary of Mathematics (2nd ed. ed.). MIT Press. p. 1141. ISBN0-262-59020-4.

5.3 See also

22

Chapter 6

Bidiagonal matrix

In mathematics, a bidiagonal matrix is a matrix with non-zero entries along the main diagonal and either the diagonalabove or the diagonal below. This means there are exactly two non zero diagonals in the matrix.When the diagonal above the main diagonal has the non-zero entries the matrix is upper bidiagonal. When thediagonal below the main diagonal has the non-zero entries the matrix is lower bidiagonal.For example, the following matrix is upper bidiagonal:

0BB@1 4 0 00 4 1 00 0 3 40 0 0 3

1CCAand the following matrix is lower bidiagonal:

0BB@1 0 0 02 4 0 00 3 3 00 0 4 3

1CCA:

6.1 UsageOne variant of the QR algorithm starts with reducing a general matrix into a bidiagonal one,[1] and the Singular valuedecomposition uses this method as well.

6.2 See also

Diagonal matrix

List of matrices

LAPACK

Bidiagonalization

Hessenberg form The Hessenberg form is similar, but has more non zero diagonal lines than 2.

Tridiagonal matrix with three diagonals

23

24 CHAPTER 6. BIDIAGONAL MATRIX

6.3 References Stewart, G. W. (2001) Matrix Algorithms, Volume II: Eigensystems. Society for Industrial and Applied Mathe-matics. ISBN 0-89871-503-2.

[1] Bochkanov Sergey Anatolyevich. ALGLIB User Guide - General Matrix operations - Singular value decomposition . AL-GLIB Project. 2010-12-11. URL:http://www.alglib.net/matrixops/general/svd.php. Accessed: 2010-12-11. (Archivedby WebCite at http://www.webcitation.org/5utO4iSnR)

6.4 External links High performance algorithms for reduction to condensed (Hessenberg, tridiagonal, bidiagonal) form

Chapter 7

Big M method

In operations research, the Big M method is a method of solving linear programming problems using the simplexalgorithm. The Big M method extends the power of the simplex algorithm to problems that contain greater-thanconstraints. It does so by associating the constraints with large negative constants which would not be part of anyoptimal solution, if it exists.

7.1 AlgorithmThe simplex algorithm is the original and still one of the most widely used methods for solving linear maximizationproblems. However, to apply it, the origin (all variables equal to 0) must be a feasible point. This condition is satisedonly when all the constraints (except non-negativity) are less-than constraints with a positive constant on the right-hand side. The Big M method introduces surplus and articial variables to convert all inequalities into that form. TheBig M refers to a large number associated with the articial variables, represented by the letter M.The steps in the algorithm are as follows:

1. Multiply the inequality constraints to ensure that the right hand side is positive.

2. If the problem is of minimization, transform to maximization by multiplying the objective by 13. For any greater-than constraints, introduce surplus and articial variables (as shown below)

4. Choose a large positive M and introduce a term in the objective of the form -M multiplying the articialvariables

5. For less-than or equal constraints, introduce slack variables so that all constraints are equalities

6. Solve the problem using the usual simplex method.

For example x + y 100 becomes x + y + s1 = 100, whilst x + y 100 becomes x + y a1 = 100. The articialvariables must be shown to be 0. The function to be maximised is rewritten to include the sum of all the articialvariables. Then row reductions are applied to gain a nal solution.The value of M must be chosen suciently large so that the articial variable would not be part of any feasiblesolution.For a suciently large M, the optimal solution contains any articial variables in the basis (i.e. positive values) if andonly if the problem is not feasible.

7.2 Other usageThe Big M method sometimes refers to any formulation of a linear optimization problem in which violations of aconstraint are associated with a large positive penalty constant, M.

25

26 CHAPTER 7. BIG M METHOD

Another usage refers to using a large positive constant, M, to ensure that the constraint is not tight. For example,suppose x and y are 0. Then for a suciently large M and z binary variable (0 or 1), the following constraintensures that when z=1, x=y: x y M(1 z)

7.3 See also Two phase method (linear programming) another approach for solving problems with >= constraints KarushKuhnTucker conditions, which apply to Non-Linear Optimization problems with inequality con-straints.

7.4 References and external linksBibliography

Griva, Igor; Nash, Stephan G.; Sofer, Ariela. Linear and Nonlinear Optimization (2nd ed.). Society for Indus-trial Mathematics. ISBN 978-0-89871-661-0.

Discussion

Simplex Big M Method, Lynn Killen, Dublin City University. The Big M Method, businessmanagementcourses.org The Big M Method, Mark Hutchinson

Chapter 8

Bilinear form

In mathematics, more specically in abstract algebra and linear algebra, a bilinear form (also informally a two-form)on a vector space V is a bilinear map V V K, where K is the eld of scalars. In other words, a bilinear form is afunction B : V V K which is linear in each argument separately:

B(u + v, w) = B(u, w) + B(v, w) B(u, v + w) = B(u, v) + B(u, w) B(u, v) = B(u, v) = B(u, v)

The denition of a bilinear form can be extended to include modules over a commutative ring, with linear mapsreplaced by module homomorphisms.When K is the eld of complex numbers C, one is often more interested in sesquilinear forms, which are similar tobilinear forms but are conjugate linear in one argument.

8.1 Coordinate representationLet V Kn be an n-dimensional vector space with basis {e1, ..., en}. Dene the n n matrix A by Aij = B(ei, ej). Ifthe n 1 matrix x represents a vector v with respect to this basis, and analogously, y represents w, then:

B(v;w) = xTAy =nX

i;j=1

aijxiyj :

Suppose {f1, ..., fn} is another basis for V, such that:

[f1, ..., fn] = [e1, ..., en]S

where S GL(n, K). Now the new matrix representation for the bilinear form is given by: STAS.

8.2 Maps to the dual spaceEvery bilinear form B on V denes a pair of linear maps from V to its dual space V. Dene B1, B2: V V by

B1(v)(w) = B(v, w)B2(v)(w) = B(w, v)

This is often denoted as

27

28 CHAPTER 8. BILINEAR FORM

B1(v) = B(v, )B2(v) = B(, v)

where the dot ( ) indicates the slot into which the argument for the resulting linear functional is to be placed.For a nite-dimensional vector space V, if either of B1 or B2 is an isomorphism, then both are, and the bilinear formB is said to be nondegenerate. More concretely, for a nite-dimensional vector space, non-degenerate means thatevery non-zero element pairs non-trivially with some other element:

B(x; y) = 0 for all y 2 V implies that x = 0 andB(x; y) = 0 for all x 2 V implies that y = 0.

The corresponding notion for a module over a ring is that a bilinear form is unimodular ifV V is an isomorphism.Given a nite-dimensional module over a commutative ring, the pairing may be injective (hence nondegenerate inthe above sense) but not unimodular. For example, over the integers, the pairing B(x, y) = 2xy is nondegenerate butnot unimodular, as the induced map from V = Z to V = Z is multiplication by 2.If V is nite-dimensional then one can identify V with its double dual V. One can then show that B2 is the transposeof the linear map B1 (if V is innite-dimensional then B2 is the transpose of B1 restricted to the image of V in V).Given B one can dene the transpose of B to be the bilinear form given by

B(v, w) = B(w, v).

The left radical and right radical of the form B are the kernels of B1 and B2 respectively;[1] they are the vectorsorthogonal to the whole space on the left and on the right.[2]

If V is nite-dimensional then the rank of B1 is equal to the rank of B2. If this number is equal to dim(V) then B1and B2 are linear isomorphisms from V to V. In this case B is nondegenerate. By the ranknullity theorem, this isequivalent to the condition that the left and equivalently right radicals be trivial. For nite-dimensional spaces, this isoften taken as the denition of nondegeneracy:

Denition: B is nondegenerate if and only if B(v, w) = 0 for all w implies v = 0.

Given any linear map A : V V one can obtain a bilinear form B on V via

B(v, w) = A(v)(w).

This form will be nondegenerate if and only if A is an isomorphism.IfV is nite-dimensional then, relative to some basis forV, a bilinear form is degenerate if and only if the determinantof the associated matrix is zero. Likewise, a nondegenerate form is one for which the determinant of the associatedmatrix is non-zero (the matrix is non-singular). These statements are independent of the chosen basis. For a moduleover a ring, a unimodular form is one for which the determinant of the associate matrix is a unit (for example 1),hence the term; note that a form whose matrix is non-zero but not a unit will be nondegenerate but not unimodular,for example B(x, y) = 2xy over the integers.

8.3 Symmetric, skew-symmetric and alternating formsWe dene a form to be

symmetric if B(v, w) = B(w, v) for all v, w in V ; alternating if B(v, v) = 0 for all v in V; skew-symmetric if B(v, w) = B(w, v) for all v, w in V ;

Proposition: Every alternating form is skew-symmetric.Proof: This can be seen by expanding B(v + w, v + w).

8.4. DERIVED QUADRATIC FORM 29

If the characteristic ofK is not 2 then the converse is also true: every skew-symmetric form is alternating. If, however,char(K) = 2 then a skew-symmetric form is the same as a symmetric form and there exist symmetric/skew-symmetricforms which are not alternating.A bilinear form is symmetric (resp. skew-symmetric) if and only if its coordinate matrix (relative to any basis)is symmetric (resp. skew-symmetric). A bilinear form is alternating if and only if its coordinate matrix is skew-symmetric and the diagonal entries are all zero (which follows from skew-symmetry when char(K) 2).A bilinear form is symmetric if and only if the maps B1, B2: V V are equal, and skew-symmetric if and only ifthey are negatives of one another. If char(K) 2 then one can decompose a bilinear form into a symmetric and askew-symmetric part as follows

B =1

2(B B)

where B is the transpose of B (dened above).

8.4 Derived quadratic formFor any bilinear form B : V V K, there exists an associated quadratic form Q : V K dened by Q : V K : v B(v,v).When char(K) 2, the quadratic formQ is determined by the symmetric part of the bilinear formB and is independentof the antisymmetric part. In this case there is a one-to-one correspondence between the symmetric part of the bilinearform and the quadratic form, and it makes sense to speak of the symmetric bilinear form associated with a quadraticform.When char(K) = 2 and dim V > 1, this correspondence between quadratic forms and symmetric bilinear forms breaksdown.

8.5 Reexivity and orthogonalityDenition: A bilinear form B : V V K is called reexive if B(v, w) = 0 implies B(w, v) = 0 for

all v, w in V.

Denition: Let B : V V K be a reexive bilinear form. v, w in V are orthogonal with respectto B if and only if B(v, w) = 0.

A form B is reexive if and only if it is either symmetric or alternating.[3] In the absence of reexivity we have todistinguish left and right orthogonality. In a reexive space the left and right radicals agree and are termed the kernelor the radical of the bilinear form: the subspace of all vectors orthogonal with every other vector. A vector v, withmatrix representation x, is in the radical of a bilinear form with matrix representation A, if and only if Ax = 0 xTA= 0. The radical is always a subspace of V. It is trivial if and only if the matrix A is nonsingular, and thus if and onlyif the bilinear form is nondegenerate.SupposeW is a subspace. Dene the orthogonal complement[4]

W? = fv j B(v;w) = 0 8w 2Wg :

For a non-degenerate form on a nite dimensional space, the map W W is bijective, and the dimension of Wis dim(V) dim(W).

8.6 Dierent spacesMuch of the theory is available for a bilinear mapping to the base eld

30 CHAPTER 8. BILINEAR FORM

B : V W K.

In this situation we still have induced linear mappings from V to W, and from W to V. It may happen that thesemappings are isomorphisms; assuming nite dimensions, if one is an isomorphism, the other must be. When thisoccurs, B is said to be a perfect pairing.In nite dimensions, this is equivalent to the pairing being nondegenerate (the spaces necessarily having the samedimensions). For modules (instead of vector spaces), just as how a nondegenerate form is weaker than a unimodularform, a nondegenerate pairing is a weaker notion than a perfect pairing. A pairing can be nondegenerate withoutbeing a perfect pairing, for instance Z Z Z via (x,y) 2xy is nondegenerate, but induces multiplication by 2 onthe map Z Z.Terminology varies in coverage of bilinear forms. For example, F. Reese Harvey discusses eight types of innerproduct.[5] To dene them he uses diagonal matrices Aij having only +1 or 1 for non-zero elements. Some of theinner products are symplectic forms and some are sesquilinear forms or Hermitian forms. Rather than a generaleld K, the instances with real numbersR, complex numbers C, and quaternionsH are spelled out. The bilinear form

pXk=1

xkyk nX

k=p+1

xkyk

is called the real symmetric case and labeled R(p, q), where p + q = n. Then he articulates the connection totraditional terminology:[6]

Some of the real symmetric cases are very important. The positive denite case R(n, 0) is called Eu-clidean space, while the case of a single minus, R(n1, 1) is called Lorentzian space. If n = 4, thenLorentzian space is also called Minkowski space or Minkowski spacetime. The special case R(p, p) willbe referred to as the split-case.

8.7 Relation to tensor productsBy the universal property of the tensor product, bilinear forms on V are in 1-to-1 correspondence with linear mapsV V K. If B is a bilinear form on V the corresponding linear map is given by

v w B(v, w)

The set of all linear maps V V K is the dual space of V V, so bilinear forms may be thought of as elements of

(V V) V V

Likewise, symmetric bilinear forms may be thought of as elements of Sym2(V) (the second symmetric power ofV), and alternating bilinear forms as elements of 2V (the second exterior power of V).

8.8 On normed vector spacesDenition: A bilinear form on a normed vector space (V, ) is bounded, if there is a constant C

such that for all u, v V,

B(u; v) Ckukkvk:

Denition: A bilinear form on a normed vector space (V, ) is elliptic, or coercive, if there is aconstant c > 0 such that for all u V,

B(u;u) ckuk2:

8.9. SEE ALSO 31

8.9 See also Bilinear map

Bilinear operator

Inner product space

Linear form

Multilinear form

Quadratic form

Positive semi denite

Sesquilinear form

8.10 Notes[1] Jacobson 2009 p.346

[2] Zhelobenko, Dmitri Petrovich (2006). Principal Structures and Methods of Representation Theory. Translations of Math-ematical Monographs. American Mathematical Society. p. 11. ISBN 0-8218-3731-1.

[3] Grove 1997

[4] Adkins & Weintraub (1992) p.359

[5] Harvey p. 22

[6] Harvey p 23

8.11 References Jacobson, Nathan (2009). Basic Algebra I (2nd ed.). ISBN 978-0-486-47189-1.

Adkins, William A.; Weintraub, Steven H. (1992). Algebra: An Approach via Module Theory. Graduate Textsin Mathematics 136. Springer-Verlag. ISBN 3-540-97839-9. Zbl 0768.00003.

Cooperstein, Bruce (2010). Ch 8: Bilinear Forms and Maps. Advanced Linear Algebra. CRC Press. pp.24988. ISBN 978-1-4398-2966-0.

Grove, Larry C. (1997). Groups and characters. Wiley-Interscience. ISBN 978-0-471-16340-4.

Halmos, Paul R. (1974). Finite-dimensional vector spaces. Undergraduate Texts in Mathematics. Berlin, NewYork: Springer-Verlag. ISBN 978-0-387-90093-3. Zbl 0288.15002.

Harvey, F. Reese (1990) Spinors and calibrations, Ch 2:The Eight Types of Inner Product Spaces, pp 1940,Academic Press, ISBN 0-12-329650-1 .

M. Hazewinkel ed. (1988) Encyclopedia of Mathematics, v.1, p. 390, Kluwer Academic Publishers

Milnor, J.; Husemoller, D. (1973). Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzge-biete 73. Springer-Verlag. ISBN 3-540-06009-X. Zbl 0292.10016.

Shilov, Georgi E. (1977). Silverman, Richard A., ed. Linear Algebra. Dover. ISBN 0-486-63518-X.

Shafarevich, I. R.; A. O. Remizov (2012). Linear Algebra and Geometry. Springer. ISBN 978-3-642-30993-9.

32 CHAPTER 8. BILINEAR FORM

8.12 External links Hazewinkel, Michiel, ed. (2001), Bilinear form, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Bilinear form at PlanetMath.org.

This article incorporates material from Unimodular on PlanetMath, which is licensed under the Creative CommonsAttribution/Share-Alike License.

Chapter 9

Binomial inverse theorem

In mathematics, the binomial inverse theorem is useful for expressing matrix inverses in dierent ways.If A, U, B, V are matrices of sizes pp, pq, qq, qp, respectively, then

(A+ UBV)1 = A1 A1UB B+ BVA1UB1 BVA1provided A and B + BVA1UB are nonsingular. Nonsingularity of the latter requires that B1 exist since it equalsB(I+VA1UB) and the rank of the latter cannot exceed the rank of B.[1]

Since B is invertible, the two B terms anking the parenthetical quantity inverse in the right-hand side can be replacedwith (B1)1, which results in

(A+ UBV)1 = A1 A1U B1 + VA1U1 VA1:This is the Woodbury matrix identity, which can also be derived using matrix blockwise inversion.A more general formula exists when B is singular and possibly even non-square:[1]

(A+ UBV)1 = A1 A1U(I+ BVA1U)1BVA1:Formulas also exist for certain cases in which A is singular.[2]

9.1 VericationFirst notice that

(A+ UBV)A1UB = UB+ UBVA1UB = UB+ BVA1UB

:

Now multiply the matrix we wish to invert by its alleged inverse:

(A+ UBV)A1 A1UB B+ BVA1UB1 BVA1

= Ip + UBVA1 UB+ BVA1UB

B+ BVA1UB

1 BVA1= Ip + UBVA1 UBVA1 = Ipwhich veries that it is the inverse.So we get that ifA1 and

B+ BVA1UB

1 exist, then (A+ UBV)1 exists and is given by the theorem above.[3]33

34 CHAPTER 9. BINOMIAL INVERSE THEOREM

9.2 Special cases

9.2.1 First

If p = q and U = V = Ip is the identity matrix, then

(A+ B)1 = A1 A1B B+ BA1B1 BA1:Remembering the identity

(CD)1 = D1C1;

we can also express the previous equation in the simpler form as

(A+ B)1 = A1 A1 I+ BA11 BA1:9.2.2 Second

If B = Iq is the identity matrix and q = 1, then U is a column vector, written u, and V is a row vector, written vT.Then the theorem implies the Sherman-Morrison formula:

A+ uvT

1= A1 A

1uvTA11 + vTA1u :

This is useful if one has a matrix A with a known inverse A1 and one needs to invert matrices of the form A+uvTquickly for various u and v.

9.2.3 Third

If we set A = Ip and B = Iq, we get

(Ip + UV)1 = Ip U (Iq + VU)1 V:

In particular, if q = 1, then

I+ uvT

1= I uv

T

1 + vTu ;

which is a particular case of the Sherman-Morrison formula given above.

9.3 See also Invertible matrix

Matrix determinant lemma

Moore-Penrose pseudoinverse#Updating the pseudoinverse

9.4. REFERENCES 35

9.4 References[1] Henderson, H. V., and Searle, S. R. (1981), On deriving the inverse of a sum of matrices, SIAM Review 23, pp. 53-60.

[2] Kurt S. Riedel, A ShermanMorrisonWoodbury Identity for Rank Augmenting Matrices with Application to Center-ing, SIAM Journal on Matrix Analysis and Applications, 13 (1992)659-662, doi:10.1137/0613040 preprint MR 1152773

[3] Gilbert Strang (2003). Introduction to Linear Algebra (3rd edition ed.). Wellesley-Cambridge Press: Wellesley, MA. ISBN0-9614088-9-8.

Chapter 10

Braket notation

In quantum mechanics, braket notation is a standard notation for describing quantum states, composed of anglebrackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics. It isso called because the inner product (or dot product on a complex vector space) of two states is denoted by

h j iconsisting of a left part, hj called the bra /br/, and a right part, j i , called the ket /kt/. The notation wasintroduced in 1939 by Paul Dirac[1] and is also known as Dirac notation, though the notation has precursors inGrassmann's use of the notation [ j ] for his inner products nearly 100 years earlier.[2][3]Braket notation is widespread in quantum mechanics: almost every phenomenon that is explained using quantummechanicsincluding a large portion of modern physics is usually explained with the help of bra-ket notation.Part of the appeal of the notation is the abstract representation-independence it encodes, together with its versatilityin producing a specic representation (e.g. x, or p, or eigenfunction base) without much ado, or excessive relianceon the nature of the linear spaces involved. The overlap expression h j i is typically interpreted as the probabilityamplitude for the state to collapse into the state .

10.1 Vector spaces

10.1.1 Background: Vector spacesMain article: Vector space

In physics, basis vectors allow any Euclidean vector to be represented geometrically using angles and lengths, indierent directions, i.e. in terms of the spatial orientations. It is simpler to see the notational equivalences betweenordinary notation and bra-ket notation; so, for now, consider a vectorA starting at the origin and ending at an elementof 3-d Euclidean space; the vector then is specied by this end-point, a triplet of elements in the eld of real numbers,symbolically dubbed as A 3.The vector A can be written using any set of basis vectors and corresponding coordinate system. Informally basisvectors are like building blocks of a vector": they are added together to compose a vector, and the coordinates arethe numerical coecients of basis vectors in each direction. Two useful representations of a vector are simply a linearcombination of basis vectors, and column matrices. Using the familiar Cartesian basis, a vector A may be written as

A := Axex +Ayey +Azez = Ax

0@100

1A+Ay0@010

1A+Az0@001

1A

=

0@Ax00

1A+0@ 0Ay

0

1A+0@ 00Az

1A =0@AxAyAz

1A36

10.1. VECTOR SPACES 37

A

A y e y

A x e x

A z

e z

z

y

x

= A e x A x

= A y e y A

= A e z z A

A + + = A x A z e y e x z e y A A + + = e x A x e y A y e z A z

= A e x A x

= A e y A y

= A e z A z

A A

z

y

x

e x A x

e y A y

e z

A z

3d real vector components and bases projection; similarities between vector calculus notation and Dirac notation. Projection is animportant feature of the Dirac notation.

respectively, where ex, ey, ez denote the Cartesian basis vectors (all are orthogonal unit vectors) and Ax, Ay, Az arethe corresponding coordinates, in the x, y, z directions. In a more general notation, for any basis in 3-d space onewrites

A := A1e1 +A2e2 +A3e3 =

0@A1A2A3

1AGeneralizing further, consider a vector A in an N-dimensional vector space over the eld of complex numbers ,symbolically stated as A N . The vector A is still conventionally represented by a linear combination of basisvectors or a column matrix:

A :=NXn=1

Anen =

0BBB@A1A2...

AN

1CCCAthough the coordinates are now all complex-valued.Even more generally, A can be a vector in a complex Hilbert space. Some Hilbert spaces, like N , have nitedimension, while others have innite dimension. In an innite-dimensional space, the column-vector representationof A would be a list of innitely many complex numbers.

10.1.2 Ket notation for vectors

Rather than boldtype, over arrows, underscores etc. conventionally used elsewhere, A; ~A; A , Diracs notation for avector uses vertical bars and angular brackets: jAi . When this notation is used, these vectors are called ket, readas ket-A.[4] This applies to all vectors, the resultant vector and the basis. The previous vectors are now written

38 CHAPTER 10. BRAKET NOTATION

jAi = Axjexi+Ayjeyi+Azjezi :=0@AxAyAz

1A;or in a more easily generalized notation,

jAi = A1je1i+A2je2i+A3je3i :=0@A1A2A3

1A;The last one may be written in short as

jAi = A1j1i+A2j2i+A3j3i :

Note how any symbols, letters, numbers, or even wordswhatever serves as a convenient labelcan be used as thelabel inside a ket. In other words, the symbol " jAi " has a specic and universal mathematical meaning, while justthe "A" by itself does not. Nevertheless, for convenience, there is usually some logical scheme behind the labelsinside kets, such as the common practice of labeling energy eigenkets in quantum mechanics through a listing of theirquantum numbers. Further note that a ket and its representation by a coordinate vector are not the same mathematicalobject: a ket does not require specication of a basis, whereas the coordinate vector needs a basis in order to be welldened (the same holds for an operator and its representation by a matrix).[5] In this context, one should best use asymbol dierent than the equal sign, for example the symbol , read as is represented by.

10.1.3 Inner products and brasMain article: Inner product

An inner product is a generalization of the dot product. The inner product of two vectors is a scalar. bra-ket notationuses a specic notation for inner products:

hAjBi = ket of product inner thejAi ket with jBi

For example, in three-dimensional complex Euclidean space,

hAjBi := AxBx +AyBy +AzBzwhere Ai denotes the complex conjugate of A. A special case is the inner product of a vector with itself, which isthe square of its norm (magnitude):

hAjAi := jAxj2 + jAyj2 + jAzj2

bra-ket notation splits this inner product (also called a bracket) into two pieces, the bra and the ket":

hAjBi = ( hAj ) ( jBi )

where hAj is called a bra, read as bra-A, and jBi is a ket as above.The purpose of splitting the inner product into a bra and a ket is that both the bra hAj and the ket jBi are meaningfulon their own, and can be used in other contexts besides within an inner product. There are two main ways to thinkabout the meanings of separate bras and kets:

10.1. VECTOR SPACES 39

Bras and kets as row and column vectors

For a nite-dimensional vector space, using a xed orthonormal basis, the inner product can be written as a matrixmultiplication of a row vector with a column vector:

hAjBi := A1B1 +A2B2 + +ANBN=A1 A

2 AN

0BBB@B1B2...

BN

1CCCABased on this, the bras and kets can be dened as:

hAj :=A1 A2 AN

jBi :=

0BBB@B1B2...

BN

1CCCAand then it is understood that a bra next to a ket implies matrix multiplication.The conjugate transpose (also called Hermitian conjugate) of a bra is the corresponding ket and vice versa:

hAjy = jAi; jAiy = hAjbecause if one starts with the bra

A1 A

2 AN

;

then performs a complex conjugation, and then a matrix transpose, one ends up with the ket

0BBB@A1A2...

AN

1CCCABras as linear operators on kets

Main articles: Dual space and Riesz representation theorem

A more abstract denition, which is equivalent but more easily generalized to innite-dimensional spaces, is to saythat bras are linear functionals on kets, i.e. operators that input a ket and output a complex number. The bra operatorsare dened to be consistent with the inner product.In mathematics terminology, the vector space of bras is the dual space to the vector space of kets, and correspondingbras and kets are related by the Riesz representation theorem.

10.1.4 Non-normalizable states and non-Hilbert spacesbra-ket notation can be used even if the vector space is not a Hilbert space.In quantum mechanics, it is common practice to write down kets which have innite norm, i.e. non-normalisablewavefunctions. Examples include states whose wavefunctions are Dirac delta functions or innite plane waves. These

40 CHAPTER 10. BRAKET NOTATION

do not, technically, belong to the Hilbert space itself. However, the denition of Hilbert space can be broadenedto accommodate these states (see the GelfandNaimarkSegal construction or rigged Hilbert spaces). The bra-ketnotation continues to work in an analogous way in this broader context.For a rigorous treatment of the Dirac inner product of non-normalizable states, see the denition given by D.Carf.[6][7] For a rigorous denition of basis with a continuous set of indices and consequently for a rigorous denitionof position and momentum basis, see.[8] For a rigorous statement of the expansion of an S-diagonalizable operator,or observable, in its eigenbasis or in another basis, see.[9]

Banach spaces are a dierent generalization of Hilbert spaces. In a Banach space B, the vectors may be notated bykets and the continuous linear functionals by bras. Over any vector space without topology, we may also notate thevectors by kets and the linear functionals by bras. In these more general contexts, the bracket does not have themeaning of an inner product, because the Riesz representation theorem does not apply.

10.2 Usage in quantum mechanicsThe mathematical structure of quantum mechanics is based in large part on linear algebra:

Wave functions and other quantum states can be represented as vectors in a complex Hilbert space. (The exactstructure of this Hilbert space depends on the situation.) In bra-ket notation, for example, an electron might bein the state |. (Technically, the quantum states are rays of vectors in the Hilbert space, as c| correspondsto the same state for any nonzero complex number c.)

Quantum superpositions can be described as vector sums of the constituent states. For example, an electron inthe state |1 + i |2 is in a quantum superposition of the states |1 and |2.

Measurements are associated with linear operators (called observables) on the Hilbert space of quantum states.

Dynamics are also described by linear operators on the Hilbert space. For example, in the Schrdinger picture,there is a linear time evolution operator U with the property that if an electron is in state | right now, then inone second it will be in the state U |, the same U for every possible |.

Wave function normalization is scaling a wave function so that its norm is 1.

Since virtually every calculation in quantum mechanics involves vectors and linear operators, it can involve, and oftendoes involve, bra-ket notation. A few examples follow:

10.2.1 Spinless positionspace wave function

5 4

Re( )

A k

A 2 A e 2 =

A 6 A e 6 =

Im(

)A

k

k 2 1 3 6

Discrete components Ak of a complex vector |A = k Ak|ek, which belongs to a countably innite-dimensionalHilbert space; there are countably innitely many k values and basis vectors |ek.

10.2. USAGE IN QUANTUM MECHANICS 41

b

Re( )

(b) = b

Im(

)a (a) =

x a

Continuous components (x) of a complex vector | = dx (x)|x, which belongs to an uncountably innite-dimensional Hilbert space; there are innitely many x values and basis vectors |x.Components of complex vectors plotted against index number; discrete k and continuous x. Two particular compo-nents out of innitely many are highlighted.

The Hilbert space of a spin0 point particle is spanned by a position basis" { |r }, where the label r extends overthe set of all points in position space. Since there are uncountably innitely many vectors in the basis, this is anuncountably innite-dimensional Hilbert space. The dimensions of the Hilbert space (usually innite) and positionspace (usually 1, 2 or 3) are not to be conated.Starting from any ket | in this Hilbert space, we can dene a complex scalar function of r, known as a wavefunction:

(r) def= hrjiOn the left side, (r) is a function mapping any point in space to a complex number; on the right side, | = d3r(r) |r is a ket.It is then customary to dene linear operators acting on wavefunctions in terms of linear operators acting on kets, by

A(r) def= hrjAji:For instance, the momentum operator p has the following form,

p(r) def= hrjpji = i~r(r):One occasionally encounters a sloppy expression like

rji;though this is something of a (common) abuse of notation. The dierential operator must be understood to be anabstract operator, acting on kets, that has the eect of dierentiating wavefunctions once the expression is projectedinto the position basis,

rhrji;even though, in the momentum basis, the operator amounts to a mere multiplication operator (by ip).

10.2.2 Overlap of statesIn quantum mechanics the expression | is typically interpreted as the probability amplitude for the state tocollapse into the state . Mathematically, this means the coecient for the projection of onto . It is also describedas the projection of state onto state .

42 CHAPTER 10. BRAKET NOTATION

10.2.3 Changing basis for a spin-1/2 particleA stationary spin- particle has a two-dimensional Hilbert space. One orthonormal basis is:

j "zi; j #zi

where j "zi is the state with a denite value of the spin operator Sz equal to +1/2 and j #zi is the state with a denitevalue of the spin operator Sz equal to 1/2.Since these are a basis, any quantum state of the particle can be expressed as a linear combination (i.e., quantumsuperposition) of these two states:

j i = a j "zi+ b j #zi

where a, b are complex numbers.A dierent basis for the same Hilbert space is:

j "xi; j #xi

dened in terms of Sx rather than Sz.Again, any state of the particle can be expressed as a linear combination of these two:

j i = c j "xi+ d j #xi

In vector form, you might write

j i :=a b

; OR j i :=

c d

depending on which basis you are using. In other words, the coordinates of a vector depend on the basis used.There is a mathematical relationship between a, b, c, d; see change of basis.

10.2.4 Misleading usesThere are a few conventions and abuses of notation that are generally accepted by the physics community, but whichmight confuse the non-initiated.It is common among physicists to use the same symbol for labels and constants in the same equation. It supposedlybecomes easier to identify that the constant is related to the labeled object, and is claimed that the divergent natureof each will eliminate any ambiguity and no further dierentiation is required. For example, | = |, where thesymbol is used simultaneously as the name of the operator , its eigenvector | and the associated eigenvalue .Something similar occurs in component notation of vectors. While (uppercase) is traditionally associated withwavefunctions, (lowercase) may be used to denote a label, a wave function or complex constant in the same context,usually dierentiated only by a subscript.The main abuses are including operations inside the vector labels. This is usually done for a fast notation of scalingvectors. E.g. if the vector | is scaled by 2, it might be denoted by |/2, which makes no sense since is a label,not a function or a number, so you can't perform operations on it.This is especially common when denoting vectors as tensor products, where part of the labels are moved outsidethe designed slot. E.g. | = |/21 |/22. Here part of the labeling that should state that all three vectors aredierent was moved outside the kets, as subscripts 1 and 2. And a further abuse occurs, since is meant to refer tothe norm of the rst vector which is a label is denoting a value.

10.3. LINEAR OPERATORS 43

10.3 Linear operatorsSee also: Linear operator

10.3.1 Linear operators acting on kets

A linear operator is a map that