Vector Spaces and SubSpaces

Linear AlgebraMATH 2076

Section 4.1.B Vector Spaces 20 February 2017 1 / 1

What is a Vector Space?

A vector space is a “bunch” of objects—that we call vectors—with theproperties that we can add any two vectors and we can multiply any vectorby any scalar.

Let V be a set. Suppose we have a way of

adding any two elements of V

multiplying any element of V by any scalar

That is,

given ~v and ~w in V, there is a ~v + ~w in V

given ~v in V and any scalar s, there is a s~v in VThen we call V a vector space, provided certain axioms hold.

Examples of Vector Spaces

Some simple examples:

1 Rn is a vector space ¨̂

2 Rm×n is the vector space of all m × n matrices (given m × n matricesA and B, we know what A + B and sA are, right?)

3 Cn is a vector space (here the coordinates are complex numbers)

4 Any vector subspace of Rn is itself a vector space, right?

5 R∞ = {(xn)∞n=1} is the vector space of all sequences (of real numbers)

Fundamental Example of a Vector Space

Let X be any set, and let F(X → R) be the family of all functions withdomain X and codomain R.

Thus each object f in F is a function X f−→ R.

Now let f and g be elements of F. Can we: add? multiply by scalars?

Let’s define f + g by the formulafor each x in X ,

(f + g

)(x) := f (x) + g(x)

and then define sf by the formulafor each x in X ,

)(x) := sf (x).

With these ways of adding and multiplying by scalars, the family

F(X → R) ={

all f with X f−→ R}

of real-valued functions on X becomes a vector space.

(f + g

)(x) := f (x) + g(x)

)(x) := sf (x).

F(X → R) ={

Now let f and g be elements of F. Can we: add?

multiply by scalars?

(f + g

)(x) := f (x) + g(x)

)(x) := sf (x).

F(X → R) ={

(f + g

)(x) := f (x) + g(x)

)(x) := sf (x).

F(X → R) ={

Let’s define f + g by the formula

for each x in X ,(f + g

)(x) := f (x) + g(x)

)(x) := sf (x).

F(X → R) ={

(f + g

)(x) := f (x) + g(x)

)(x) := sf (x).

F(X → R) ={

(f + g

)(x) := f (x) + g(x)

and then define sf by the formula

for each x in X ,(sf

)(x) := sf (x).

F(X → R) ={

(f + g

)(x) := f (x) + g(x)

)(x) := sf (x).

F(X → R) ={

(f + g

)(x) := f (x) + g(x)

)(x) := sf (x).

With these ways of adding and multiplying by scalars,

the family

F(X → R) ={

(f + g

)(x) := f (x) + g(x)

)(x) := sf (x).

F(X → R) ={

What is a SubSpace of a Vector Space?

Let V be a vector space.

We call W a vector subspace of V if and only if

1 ~0 is in W,

2 W closed wrt vector addition,

(~u, ~v in W =⇒ ~u + ~v in W),

3 W closed wrt scalar mult

(s scalar , ~v in W =⇒ s~v in W)

Note that

(3) says that whenever ~w is in W, Span{~w} lies in W, and

(2) and (3) together say that any LC of vectors in W is in W.

Basic Fact about Vector SubSpaces

Let W be a vector subspace of a vector space V. Suppose ~w1, ~w2, . . . , ~wp

are in W. Then Span{~w1, ~w2, . . . , ~wp} lies in W.

Homework: Explain why this fact is true.

Let V be a vector space. We call W a vector subspace of V if and only if

1 ~0 is in W,

(~u, ~v in W =⇒ ~u + ~v in W),

Note that

1 ~0 is in W,

(~u, ~v in W =⇒ ~u + ~v in W),

Note that

1 ~0 is in W,

(~u, ~v in W =⇒ ~u + ~v in W),

Note that

1 ~0 is in W,

2 W closed wrt vector addition, (~u, ~v in W =⇒ ~u + ~v in W),

Note that

1 ~0 is in W,

Note that

1 ~0 is in W,

3 W closed wrt scalar mult (s scalar , ~v in W =⇒ s~v in W).

Note that

1 ~0 is in W,

Note that

1 ~0 is in W,

Note that

1 ~0 is in W,

Note that

1 ~0 is in W,

Note that

1 ~0 is in W,

Note that

Examples of Vector SubSpaces of a Vector Space?

Let V be a vector space.

Here are some simple examples of vectorsubspaces of V.

W = {~0} is the trivial vector subspace

W = V is a vector subspace of itself (also kinda trivial)

W = Span{~v} (for any ~v in V)

W = Span{~v1, ~v2, . . . , ~vp} (for any ~v1, ~v2, . . . , ~vp in V)

In the last example above, we call W the subspace spanned by~v1, ~v2, . . . , ~vp, and then {~v1, ~v2, . . . , ~vp} is called a spanning set for W.

Example (Basic Vector SubSpace)

For any ~v1, ~v2, . . . , ~vp in a vector space V, Span{~v1, ~v2, . . . , ~vp} is thevector subspace of V spanned by ~v1, ~v2, . . . , ~vp.

Let V be a vector space. Here are some simple examples of vectorsubspaces of V.

Vector Subspaces of Rm×n

Recall that Rm×n is the vector space of all m × n matrices.

Which of thefollowing are vectors subspaces of Rn×n?

If not, why not?

All upper triangular n × n matrices.

All invertible n × n matrices.

All n × n matrices A with det(A) = 1.

All n × n matrices A with AT = A (the symmetric matrices).

All n × n matrices A with AT = −A (the skew-symmetric matrices).

Can you find spanning sets for the vector subspaces?

Recall that Rm×n is the vector space of all m × n matrices. Which of thefollowing are vectors subspaces of Rn×n?

If not, why not?

Recall that Rm×n is the vector space of all m × n matrices. Which of thefollowing are vectors subspaces of Rn×n? If not, why not?

Function Spaces

Let F = F(R→ R) = {all f with R f−→ R}. This is a vector space with theusual ways of adding and multiplying by scalars:

f + g is defined by

for each t in R,(f + g

)(t) := f (t) + g(t)

and sf is defined by

for each t in R,(sf

)(t) := sf (t).

Notice that we use t for our variable instead of x .

Here are some important vector subspaces of F.

{all continuous f in F}.{all differentiable f in F}.{all twice differentiable f in F}.Span{et}

(the solution set to y ′ = y)

Span{et , e−t}

(the solution set to y ′′ = y)

Span{cos(t), sin(t)}

(the solution set to y ′′ + y = 0)

Function Spaces

f + g is defined by

for each t in R,(f + g

)(t) := f (t) + g(t)

for each t in R,(sf

)(t) := sf (t).

Span{et , e−t}

Function Spaces

f + g is defined byfor each t in R,

(f + g

)(t) := f (t) + g(t)

for each t in R,(sf

)(t) := sf (t).

Span{et , e−t}

Function Spaces

(f + g

)(t) := f (t) + g(t)

for each t in R,(sf

)(t) := sf (t).

Span{et , e−t}

Function Spaces

(f + g

)(t) := f (t) + g(t)

and sf is defined byfor each t in R,

)(t) := sf (t).

Span{et , e−t}

Function Spaces

(f + g

)(t) := f (t) + g(t)

)(t) := sf (t).

Span{et , e−t}

Function Spaces

(f + g

)(t) := f (t) + g(t)

)(t) := sf (t).

Span{et , e−t}

Function Spaces

(f + g

)(t) := f (t) + g(t)

)(t) := sf (t).

{all continuous f in F}.

{all differentiable f in F}.{all twice differentiable f in F}.Span{et}

Span{et , e−t}

Function Spaces

(f + g

)(t) := f (t) + g(t)

)(t) := sf (t).

{all continuous f in F}.{all differentiable f in F}.

{all twice differentiable f in F}.Span{et}

Span{et , e−t}

Function Spaces

(f + g

)(t) := f (t) + g(t)

)(t) := sf (t).

{all continuous f in F}.{all differentiable f in F}.{all twice differentiable f in F}.

Span{et}

Span{et , e−t}

Function Spaces

(f + g

)(t) := f (t) + g(t)

)(t) := sf (t).

Span{et , e−t}

Function Spaces

(f + g

)(t) := f (t) + g(t)

)(t) := sf (t).

{all continuous f in F}.{all differentiable f in F}.{all twice differentiable f in F}.Span{et} (the solution set to y ′ = y)

Span{et , e−t}

Function Spaces

(f + g

)(t) := f (t) + g(t)

)(t) := sf (t).

Span{et , e−t}

Function Spaces

(f + g

)(t) := f (t) + g(t)

)(t) := sf (t).

Span{et , e−t} (the solution set to y ′′ = y)

Function Spaces

(f + g

)(t) := f (t) + g(t)

)(t) := sf (t).

Function Spaces

(f + g

)(t) := f (t) + g(t)

)(t) := sf (t).

Span{cos(t), sin(t)} (the solution set to y ′′ + y = 0)

The Space of all Polynomials

Let P be the family of all polynomials.

Thus p is in P if and only if p is afunction of the form

p(t) = c0 + c1t + c2t2 + · · ·+ cnt

Here c0, c1, . . . , cn are constants, called the coefficients of the polynomialp, and when cn 6= 0 we say that p has degree n. (By definition, the zeropolynomial has degree zero.)

Clearly the sum p + q of two polys p and q is again a poly, as is anyscalar multiple sp. Thus, P is a vector subspace of F.

Of course, P is also a vector space all by itself.

Let P be the family of all polynomials. Thus p is in P if and only if p is afunction of the form

p(t) = c0 + c1t + c2t2 + · · ·+ cnt

Clearly the sum p + q of two polys p and q is again a poly, as is anyscalar multiple sp.

Thus, P is a vector subspace of F.

p(t) = c0 + c1t + c2t2 + · · ·+ cnt

Subspaces of Polynomials

Let P be the family of all polynomials.

Which of the following are vectorsubspaces of P? If not, why not?

All polynomials of degree n.

All polynomials of degree n or less.

(We call this subspace Pn.)

All polynomials of even degree.

All polynomials of odd degree.

All polynomials p with p(0) = 1.

Let P be the family of all polynomials. Which of the following are vectorsubspaces of P? If not, why not?

All polynomials of degree n or less. (We call this subspace Pn.)

Vector Spaces and SubSpaces - UC...

Transcript of Vector Spaces and SubSpaces - UC...

Vector Spaces and SubSpaces - UC...

Documents

Transcript of Vector Spaces and SubSpaces - UC...

Scenic Hills Elementary School 2014 Annual Title I Parent Meeting Presented by: Herronda Harley, Professional Learning Coach September 25, 2014.

Vector Autoregression and Vector Error-Correction Models

20 30 40 50 LEICA VECTOR USMC VECTOR 21 LEICA VECTOR · 20 30 40 50 LEICA VECTOR™ USMC VECTOR 21. LEICA VECTOR. ... for Rockwell PLGR+96 / PLGR II ... pid succession ...

Vector spaces, vector algebras, and vector geometries

INTRODUCTION TO DIFFERENTIAL GEOMETRYhomepages.uc.edu/~herronda/Diff_Geometry/DiffGeom.pdf · Thus in diﬀerential geometry our spaces are equipped with an additional structure,

Vector Processing. Vector Processors Combine vector operands (inputs) element by element to produce an output vector. Typical array-oriented operations.

Vector Weighting Approach and Vector Space Decoupling ...

Vector Analysis Vector Algebra – Subtraction ...eie.polyu.edu.hk/~em/em06pdf/Vector Analysis.pdf · Vector.1 Vector Analysis Vector Algebra – Addition – Subtraction – Multiplication

Vector Es Vector Es

3. VECTOR ANALYSIS - Sonoma State University · VECTOR ANALYSIS. Chapter 3: Vector Analysis. Laws of Vector Algebra. Properties of Vector Operations. Equality of Two Vectors Commutative

Distance Vector Routing Protocols Distance Vector Routing.

An Introduction to Vectors, Vector Operators and Vector ...dl.booktolearn.com/...Introduction_to_Vectors_ab15.pdf · An Introduction to Vectors, Vector Operators and Vector Analysis

Vector and Vector Resolution. Scalar Vector Vectors.

Vector and Vector Resolution

Smart innovation! Laser air+ - Beamers & Touchscreens.nl

Vector Space Scoring - ics.uci.edulopes/teaching/cs221W12/... · Vector Space Model Vector Space Scoring • Define: Vector Space Model • A document, d, is defined as a vector:

Viruses as vector, binary, shuttle vector

zA Vector Calculus - University of Houstoncourses.egr.uh.edu/ECE/ECE3317/SectionJackson/Class...Vector Calculus. 1. ECE 3317 ... The vector Laplacian of a vector function is a vector

Signal vector spaceSignal vector space

Review: Analysis vector. VECTOR ANALYSIS 1.1SCALARS AND VECTORS 1.2VECTOR COMPONENTS AND UNIT VECTOR 1.3VECTOR ALGEBRA 1.4POSITION AND DISTANCE VECTOR.