MTH 102A Linear Algebra and Ordinary Differential...

MTH 102A

Linear Algebra and Ordinary Differential

Equations

2015-16-II Semester

Arbind Kumar Lal1

January 1, 2020

1Indian Institute of Technology Kanpur

Course Number: MTH102A

Course Name: Linear Algebra & Differential Equations.

Instructors

Arbind K Lal Off-Phone: 7662. Email: [email protected]

T Muthukumar Off-Phone: 7911. Email: [email protected]

Home Page: http : //home.iitk.ac .in/ arlal/mth102a.htm

http : //home.iitk.ac .in/ arlal/book/nptel/pdf /booklinear .pdf

Reference Books:

Advanced Engineering Mathematics - Erwin Kreyszig

Linear Algebra - Kenneth M Hoffman and Ray Kunze

• Attendance: 10 MarksAny type of Leave, whether Informed, Uninformed,Emergency, Medical , SUGC Approved, ... will betreated as ABSENCE.

Exam Due About Marks

Quiz1 ∼ week 4 LA 10

MidSem LA 30

Quiz2 ∼ week 11 DE 10

Endsem LA,DE 05, 35

Chapters Classes

Matrices and Linear Equations 4

Vector Spaces 4

Inner Product Spaces 4

Eigenvalues and Eigenvectors 6

Linear Transformations 3

Expected DateQuiz - I: 6:00 PM, January 31, Friday, 2020

• A rectangular array of numbers is called a matrix. Horizontal

arrays are called its rows; Vertical arrays are called its columns.

• An m × n matrix A = [aij ] is an array of m rows and n columns.

A =

a11 a12 . . . a1n

a21 a22 . . . a2n...

......

...

am1 am2 . . . amn

A[:, 2]- Coulmn

A[1, :]- Row

• Here m × n is called the size or the order of A.

• (i , j)-th entry of A→ aij or (A)i ,j . NOTE aij is the entry at the

intersection of the ith row and jth column.

• Mm×n(F) , Mn(F)

• R : Real numbers; C : Complex numbers

• Row vector : a matrix of single row, size 1× n.

• Column vector : a matrix of single column, size m × 1.

• Elements of Rn (or Cn) will be seen as column vectors even though

sometimes written otherwise to save space.

• A = [aij ] and B = [bij ] are equal if their order are equal and

aij = bij for each i and j

• System of linear equations

{2x + 3y = 5

3x + 2y = 5

}can be

represented as[2 3 : 5

3 2 : 5

]or

[2 3 5

3 2 5

]or

[2 3 5

3 2 5

].

• Zero Matrix: Notation :- 0 For example,

02×2 =

[0 0

0 0

]and 02×3 =

[0 0 0

0 0 0

].

• Square Matrix: Notation :- An×n or An

• Diagonal /Principal Diagonal entries: The entries

a11, a22, . . . , ann of An = [aij ], square matrix

Diagonal Matrix: Notation - diag(a11, . . . , ann), a square matrix.

aij = 0 for all i 6= j .

• Identity Matrix: Notation :- In = diag(1, . . . , 1).

αI ← Scalar matrix

• Transpose of Am×n = [aij ]: Notation :- AT , a matrix

Bn×m = [bij ], with bij = aji .

• Example: If A =

[1 4 5

0 1 2

]then, AT =

1 0

4 1

5 2

.(Row vector)T = Column vector and vice-versa.

• Thm: For any matrix A, (AT )T = A.

• Triangular Matrix: An×n = [aij ]. Then, A is upper triangular if

aij = 0, for all 1 ≤ j < i ≤ n.

∗ ∗ ∗0 ∗ ∗0 0 ∗

.

lower triangular = upper triangularT

• Conjugate transpose of Am×n = [aij ]: Notation :- A∗, a matrix

Bn×m = [bij ], with bij = aji . (A∗)∗ = A.

• Addition of Am×n = [aij ] and Bm×n = [bij ]: A + B is a matrix

Cm×n = [cij ] with cij = aij + bij .

If A =

[1 4 5

0 0 2

]and B =

[2 30 3

4 2 −6

]then,

A + B =

[3 34 8

4 2 −4

].

• Scalar multiplication of A = [aij ] with k ∈ R: kA = [kaij ].

If A =

[1 4 5

0 1 2

]and k = 5⇒ 5A =

[5 20 25

0 5 10

].

• Thm Let A,B and C be m × n matrices and k ∈ R. Then,

• A + B = B + A (commutativity).

• (A + B) + C = A + (B + C ) (associativity).

• k(`A) = (k`)A.

• (k + `)A = kA + `A.

• Let A be an m × n matrix.

• Then there exists Bm×n with A + B = 0m×n.

B is called the additive inverse of A, denoted −A = (−1)A.

• A + 0m×n = 0m×n + A = A.

The matrix 0m×n is called the additive identity.

• Matrix Multiplication / Product : Am×nBn×p = Cm×p = [cij ]

with

cij =n∑

k=1

aikbkj = ai1b1j + ai2b2j + · · ·+ ainbnj .

• Let e1 = (1, 0, . . . , 0)T ∈ Rn, e2 = (0, 1, 0, . . . , 0)T ∈ Rn. Then

eT1 e1 =[1 0 · · · 0

]

1

0...

0

= 1, whereas

e1eT1 =

1

0...

0

[1 0 · · · 0

]=

1 0 · · · 0

0 0 · · · 0...

......

0 0 · · · 0

→ called e11

• If ei ∈ Rn is the vector with 1 at the i-th place and 0, elsewhere.

Then , eT1 e2 = 0 and in general, eTi ej = δij =

{1 if i = j

0 otherwise.

• What is eieTj ?

0

0

1

0

[0 1 0

]=

0 0 0

0 0 0

0 1 0

0 0 0

→ called e32.

• How does

1 0 0

0 2 0

0 0 1

behave?

•

1 0 0

0 2 0

0 0 1

a b c d

α β γ δ

u v w z

=

a b c d

2α 2β 2γ 2δ

u v w z

.• The second row of A is multiplied with 2.

•[a b c

u v w

]1 0 0

0 2 0

0 0 1

=

[a 2b c

u 2v w

].

• The second column of A is multiplied with 2.

• If A =

[1 2 3

0 1 −1

]and B =

a b c

e f g

u v w

then,

AB =

[a + 2e + 3u b + 2f + 3v c + 2g + 3w

e − u f − v g − w

].

(AB)[1, :] = A[1, :]B

= 1[a, b, c] + 2[e, f , g ] + 3[u, v ,w ]

= [a + 2e + 3u, b + 2f + 3v , c + 2g + 3w ]

• AB[:, 1] =

[1

0

]a +

[2

1

]e +

[3

−1

]u =

[a + 2e + 3u

e − u

].

• AB[:, 2] =

[1

0

]b +

[2

1

]f +

[3

−1

]v =

[b + 2f + 3v

f − v

].

• AB[:, 3] =

[1

0

]c +

[2

1

]g +

[3

−1

]w =

[c + 2g + 3w

g − w

].

• In general,

(AB)[i , :] = ai1B[1, :] + ai2B[2, :] + · · ·+ ainB[n, :].

(AB)[:, j ] = A[:, 1]b1j + A[:, 2]b2j + · · ·+ A[:, n]bnj .

• The product AB corresponds to operating on the

1. rows of the matrix B using the entries of A.

2. columns of the matrix A using the entries of B.

• Let Am×n and Bn×p. Then,

1. AB = [A(B[:, 1]), A(B[:, 2]), . . . , A(B[:, p])].

2. Also AB =

A[1, :]B

A[2, :]B...

A[n, :]B

.

• The product Am×nBn×p is defined but, the product BA is not

defined, unless p = m.

• If Am×n and Bn×m then, AB and BA are defined but their sizes are

different.

• Even if A and B are n× n, AB need not be equal to BA. Example,[1 1

1 1

][1 0

−1 0

]=

[0 0

0 0

]6=

[1 1

−1 −1

]=

[1 0

−1 0

][1 1

1 1

].

• Let Am×n, Bn×p and Cp×r and k ∈ R.

1. (AB)C = A(BC ). Matrix multiplication is associative.

2. For any k ∈ R, (kA)B = k(AB) = A(kB).

3. A(B +C ) = AB +AC . Multiplication distributes over addition.

4. If m = n and D = diag(d1, d2, . . . , dn) then,

• AIn = InA = A. In is the Multiplicative Identity

• the first row of DA is d1 times the first row of A

• for 1 ≤ i ≤ n, the ith row of DA is di times the ith row of A.

A similar statement holds for the columns of A when A is

multiplied on the right by D.

• Inverse: Let A be a square matrix. Then, B is said to be an

inverse of A if AB = BA = I .

• Inverse of A↔ A−1. Note that

1. (A−1)−1 = A

2. (A−1)T = (AT )−1.

3. (A−1)∗ = (A∗)−1.

• Let A =

[a b

c d

]. If ad − bc 6= 0 then A−1 = 1

ad−bc

[d −b−c a

].

Ques: Let A and B be invertible. Is AB invertible?

Yes. (AB)−1 = B−1A−1.

• Ques: I have a matrix with a zero row. Is it invertible?

No. If A[i , :] is zero, then (AB)[i , :] = 0 for any B. So, AB 6= I , as

I does not have a zero row.

•

0 · · · 0

∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗

=

∗ ∗ ∗0 · · · 0

∗ ∗ ∗

.

• Ques: I have a matrix with a zero column. Is it invertible?

No. If A(:, i) is zero, then (BA)[:, i ] = 0 for any B. So, no BA will

be I , as I does not have a zero column.

•

∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗

0...

0

=

∗ 0 ∗

∗... ∗

∗ 0 ∗

• The linear system

2x +3y +z = 6

x +2y +z = 4

x −z = 0

is same as

2

1

1

x +

3

2

0

y +

1

1

−1

z =

6

4

0

or equivalently, if A =

2 3 1

1 2 1

1 0 −1

then A

x

y

z

=

6

4

0

.

• Define f : R3 → R3 by f (x) = Ax. Then,

f (e1) = Ae1 =

2

1

1

, f (e2) =

3

2

0

and f (e3) =

1

1

−1

.

• Ques: The above system asks?

1. “Does Image(f ) contain

6

4

0

”?

2. “Is

6

4

0

‘linear combination’ of

2

1

1

,

3

2

0

and

1

1

−1

”?

• Submatrix: A matrix obtained by deleting some of the rows

and/or columns of a matrix is called a submatrix of the given

matrix.

For example, if A =

[1 4 5

0 1 2

], a few submatrices of A are [1],

[2],

[1

0

], [1 5],

[1 5

0 2

], A.

Not a submatrix of A:

[1

1

],

[4

0

],

[4

2

],

[1 4

1 0

]and

[1 4

0 2

].

• Thm: Let A = [aij ] = [Pm×n Qm×r ] and B = [bij ] =

[Hn×t

Kr×t

].

Then, the size/order of AB is m × t and

AB = PH + QK .

• Example: Let A =

[1 2 0

2 5 0

]and B =

a b

c d

e f

. Then,

AB =

[1 2

2 5

][a b

c d

]+

[0

0

][e f ] =

[a + 2c b + 2d

2a + 5c 2b + 5d

].

MTH 102A Linear Algebra and Ordinary Differential...

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