MTH 102: Linear Algebra
Department of Mathematics and Statistics
Indian Institute of Technology - Kanpur
Problem Set 2
Problems marked (T) are for discussions in Tutorial sessions.
1. (T) A square matrix P is called a permutation matrix if each row and co
MTH 102
Linear Algebra
Lecture 20
Eigen Values and Eigen Vectors
T
Symmetric Matrices ( A = A )
For an n n real symmetric matrix A , its n real
eigen-vectors can be chosen to be ORTHONORMAL.
How?
Eigen Values and Eigen Vectors
T
Symmetric Matrices ( A = A
MTH 102
Linear Algebra
Lecture 18
Eigen Values and Eigen Vectors
Solve
A x = x !
Example 1: istinct Eigen Values
D
A=
4
1
1
4
1 = 3, 2 = 5
-1
x1 = c
1
1
x2 = d
1
Example 2: Repeated Eigen Values Full Eigen Vectors
3 -2 0
1 = 1, 2 = 5, 31= 5
-1
0
0=
A = -
MTH 102
Linear Algebra
Lecture 6
Vector Spaces and Subspaces
Example : Subspaces from matrices
1
A = 2
3
Columns are in
4
5
6
R3 ?
Vector Spaces and Subspaces
Example : Subspaces from matrices
1
4
c1 2 + c2 5 : c1 , c2 R
3
6
14
all linear combination of
1
Orthogonal Subspaces
Denition 1.1 A set A is said to be orthonormal if it is orthogonal and u
1 u V .
=
u
Remark: Let A be an orthogonal set of non-zero vectors then cfw_ u : u A is an
orthonormal set. So, from Gram-Schimdt orthogonalization process we
MTH 102: Linear Algebra
Department of Mathematics and Statistics
Indian Institute of Technology - Kanpur
February 20, 2013
Time: 2 hours
Maximum Score: 60
Mid-Semester Examination
INSTRUCTIONS
i. Please write your Name, Roll Number and Section correctly o
MTH 102: Linear Algebra
Department of Mathematics and Statistics
Indian Institute of Technology - Kanpur
February 2, 2013
Time: 25 minutes
Maximum Score: 25
Quiz - Set B
Name
Roll Number
1. Given that
Solution:
Section
1
2
A
1
8
1
2
A
1
8
2
5
2
2
3
3
0
1
MTH 102: Linear Algebra
Department of Mathematics and Statistics
Indian Institute of Technology - Kanpur
February 2, 2013
Time: 25 minutes
Maximum Score: 25
Quiz - Set A
Name
Roll Number
Section
1. Given that
Solution:
1
2
1
8
1
2
1
8
2
5
2
2
3
3
0
1
2
5
MTH102: ODE-Assignment-I
1.T Verify that y = 1/(x + c) is general solution of y = y 2 . Find particular solutions such
that (i) y(0) = 1, and (ii) y(0) = 1. In both the cases, nd the largest interval I on
which y is dened.
2.D Consider the dierential equa
MTH102: Assignment-2
1.D Solve the following linear rst order dierential equations using the method of variation
of parameters:
(i) xy 2y = x4
(ii) y + (cos x)y = sin x cos x
[Method: To solve y + P (x)y = R(x), rst nd the general solution yh (x) = cy1 (x
Department of Mathematics
MTL 100: Calculus
Tutorial Sheet 2: Differential Calculus in One Variable
1. Do lim x sin
x0
1
and lim x sin x exist? If yes, nd the limit; if not justify.
x
x
2. Prove that the function
1 if x Q;
0 if x Q
/
f (x) =
is discontinu
Department of Mathematics
MAL 100: Calculus
Tutorial Sheet 3: Differential Calculus in Several Variables
1. Using denition of limits, prove that
10xy 2
= 0.
(x,y)(0,0) x2 + y 2
lim
2. Show that fxy (0, 0) = fyx (0, 0) for the following function:
f (x, y)
Department of Mathematics
MTL 100: Calculus
Tutorial Sheet 1: The Real Number System, Sequences & Series
1. If x and y are arbitrary real numbers, x < y, prove that there exists at least one rational
number r satisfying x < r < y.
n=1
2. Use the Archimedi
Department of Mathematics
MAL 100: Calculus
Tutorial Sheet 4: The Definite Integral
1. Using the denition of Riemann integration, determine which of the following functions are
integrable:
(a) f (x) = [x] in [0, 3];
(b)
f (x) =
x + x2 x is rational;
x2 +
Department of Mathematics
MAL 100: Calculus
Tutorial Sheet 5: Integral Calculus in several variables
9
1. Sketch the domain D of integration corresponding to
1
3
xey dxdy. Change the order of
y
the integration and then evaluate.
f (x, y) dA where f (x, y)
Department of Mathematics
MAL 100: Calculus
Tutorial Sheet 6: Vector Calculus
1. A vector v is said to be irrotational if curl v = 0. Find constants a, b, c such that A =
( x + 2y + az)i + (bx 3y z) j + (4x + 3y + 2z)k is irrotational. Also show that A is
MTH 102
Linear Algebra
Lecture 16
Determinant
Formula
For an
n n matrix A,
+1 even # of exchanges
1 odd # of exchanges
det(A) =
(sgn )[A]1(1) [A]2(2) [A]n(n)
Sn
Sn = the set of all permutations of cfw_1, 2, . . . , n
Determinant
Formula
For an 3 3 matrix
MTH 102
Linear Algebra
Lecture 2
System of Linear Equations
Definition:
Equivalent Linear Systems
Two systems of linear equations are
equivalent if their solution sets are equal.
System of Linear Equations
A 2 by 2 example
x1 3x2 = 0
x1 + x2 = 2
Easy to
MTH 102
Linear Algebra
Lecture 4
Agenda
Matrix-Matrix Multiplication
Elementary Matrices
Inverse of a Matrix
More Matrix Operations
Some Special Matrices
Inverse of a (square) Matrix
Definition
We say that an n n matrix A is invertible if there
exists a m
MTH 102
Linear Algebra
Lecture 8
Agenda
Vector Spaces and Subspaces
Null Space of a Matrix
Solvability and Solution
Linear Independence
Span, Basis and Dimension
Solvability and Solution
Solving
Ax = b !
Solvability and Solution
Solving
Ax = b !
112 3
A =
MTH 102
Linear Algebra
Lecture 10
Agenda
Solution and Solvability
Independence, Basis, Dimension
Fundamental Subspaces
Orthogonal Vectors / Subspaces
Projections
Fundamental Subspaces
For an m n matrix A , I2
C (A) : Column Space of A I2
N (A) : Null Spac
MTH 102
Linear Algebra
Lecture 5
Agenda
More Matrix Operations
Some Special Matrices
Vector Spaces and Subspaces
Null Space of a Matrix
Solvability and Solution
Vector Spaces and Subspaces
Example
2
R=
x1
: x 1 R , x2 R
x2
Vector Spaces and Subspaces
Exam
MTH 102
Linear Algebra
Lecture 9
Agenda
Null Space of a Matrix
Solution and Solvability
Linear Independence
Span, Basis and Dimension
Fundamental Subspaces
Linear Independence
Definition
A sequence of vectors v1 , v2 , . . . , vn is LINEARLY
INDEPENDENT i
MTH 102
Linear Algebra
Lecture 15
Determinant
More Properties
P7: Determinant of a matrix is ZERO if and
only if it is SINGULAR.
E1 . . . Es A = R
det(E1 ) . . . det(Es )det(A) = det(R)
det(A) = 0 det(R) = 0
Determinant
More Properties
P8: Determinant of
MTH 102
Linear Algebra
Lecture 14
Agenda
Least Squares
Gram-Schmidt
Determinant
Inverse and Cramers Rule
Eigen Values and Eigen Vectors
Determinant
A number associated to every
SQUARE matrix Notation:
det(A)
or
det : M (n, R) R
How do we define this ?
Is
MTH 102
Linear Algebra
Lecture 12
Projections
Problem
a and a vector b, find p
p span(cfw_a)
(b p) a
Given a vector
1.
2.
b
p = xa
such that
a
Projections
Problem
a and a vector b, find p
p span(cfw_a)
(b p) a
Given a vector
1.
2.
a (b p) = 0
aT b
x= T
aa
MTH 102
Linear Algebra
Lecture 11
Fundamental Subspaces
Relations
r
nr
R
N ( A)
C ( AT )
r
n
C ( A)
N ( AT )
What does
this mean?
mr
R
m
Fundamental Subspaces
Relations
nr
Ax = b is solvable if b is in C (A).
T
T
T
If y A = 0 then, y b = 0 .
r
m
R
N ( A)
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MTH102: ODE-Assignment-I
1.P Consider the differential equations y 0 = y, x > 0, where is a constant. Show that
(i) if (x) is any solution and (x) = (x)ex , then (x) is a constant;
(ii) if < 0, then every solution tends to zero as x .
ax
+
by
+
m
, ad bc