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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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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
• 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
• 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
• 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
].