Problem Set 1
Textbook: G. Strang Linear Algebra and its applications, 4th edition
Problem set
Problem number
1.2
5,7,8,11,14,17
1.3
1,3,10,12,15,22,30
1.4
6,7,10,11,13,15,16,22,26,27,31,38,40,43
1.5
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Sc
A Short Course on
LINEAR ALGEBRA
and its
APPLICATIONS
M.Thamban Nair
Department of Mathematics
Indian Institute of Technology Madras
Contents
Preface
iv
1 Vector Spaces
1.1 Introduction . . . . . . .
CHAPTER
2
American mathematician Paul
Halmos (19162006), who in 1942
published the rst modern linear
algebra book. The title of
Halmoss book was the same as the
title of this chapter.
Finite-Dimension
System of Linear Equations
Numerical
N
i l solution
l i off differential
diff
i l equations
i
(Finite Difference Method).
Numerical solution of integral equations (Finite
Element Method, Method of Mom
INDIAN INSTITUTE OF TECHNOLOGY MADRAS
SECOND SEMESTER - 2016-2017
COURSE HANDOUT
MA2030 Linear Algebra and Numerical Analysis
Instructors : Dr. A. K. B. CHAND
Dr. PRANAV HARIDAS
Office : HSB-254 H
Off
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Module 3
1
More on Vector Spaces and Linear transformations
1.1 Let X be a vector space over the field F and given a set I, let us write
x
X I := I
X is a f unction.
For x X I , we shall write x(i)
Module 7
1
Generalized Inverse
1. Theorem Given an m n matrix A over C, there exist unitary matrices U and V of order m and n
respectively such that A = U DV where
(i)
D = diag[1 , . . . , r , 0, 0, .
Module 6
1
Inner Product Space
1. Recall that X is called a vector space over a field F when X is an abelian group equipped with an
(x,y)x
action X F X satisfying expected identities like (x + u) = x
Module 6
1
Orthogonal and Orthonormal Basis
s
1. Let X be a vector space over K, K = R or C. We note that R C and the conjugation. Let XX
K
be a sesquilinear form which is hermitian; that is, s(x, u)
Problem Set 3
Textbook: G. Strang Linear Algebra and its applications, 4th edition
Problem Set
Problem Numbers
3.3
3, 6, 10, 12, 14, 18, 22
3.4
3, 4, 6, 7, 9, 11, 13, 15, 16, 29, 30
Textbook: Edgar G.
Problem Set 4
Textbook: G. Strang Linear Algebra and its applications, 4th edition
Problem Set
Problem Numbers
4.2
14, 16, 21
4.3
23, 30, 34
4.4
1, 5
5.1
6, 9, 16, 20
5.6
10, 11, 12
Textbook: Edgar G.
CS6690 - Pattern Recognition
Tutorial 1
1. When an unbiased coin is tossed 100 times, what is the probability of
getting exactly 50 heads?
2. On an average, how many times a dice has to be rolled to g
Problem Set 6
Textbook: Probability and random processes by Geoffrey Grimmett and David
Stirzaker
Section
Problem Numbers
1.2
4,5
1.3
3,5,6,7
Source: ECE 313 UIUC course lecture notes
Section
Problem
IITM-CS5800 : Advanced Data Structures and Algorithms
PS #2
Released on Sep 6, 2017
Divide and Conquer
1. Can you modify the choice of the pivot for the QuickSort algorithm to make it run in
O(n log n
IITM-CS5800-2017 : Advanced Data Structures and Algorithms (Section 2)
Practice Problem Sheet #3
Released On : Sep 29
1. Prove that, in a matroid, the independent sets which are of maximum in size, mu
IITM-CS5800 : Advanced Data Structures and Algorithms
Problem Set #1
Released on August 15, 2017
1. Given an array A[1, . . . , n] of n distinct integers, the goal is to count the number of
inversions
Problem Set 5
Problems on topics related to eigenvalues and eigenvectors
Textbook: G. Strang Linear Algebra and its applications, 4th edition
Problem Set
Problem Numbers
5.2
2, 7, 16, 34, 36
5.3
1, 3,
CS6690 - Pattern Recognition
Tutorial 2
1. Find the eigen value and eigen vectors of
1 0
A = 0 1
0 0
3
B = 2
3
C=
1
0
the following matrices:
0
0
1
6 3
4 3
6 4
2
1
2. Show that if A is positive defini
Problem Set 2
Textbook: G. Strang Linear Algebra and its applications, 4th edition
Problem set
Problem numbers
2.1
5, 8, 10, 22, 23, 24, 28
2.2
1, 2, 4, 8, 10, 27, 29, 33, 39, 42, 43, 56, 60,
63, 66,
IITM-CS5800 : Advanced Data Structures and Algorithms
Extra Problem Set #1
Released on Sept 8, 2017
1. Compare the following using O, , and notations.
(a) T (n) = n! and f (n) = 2n log n .
(b) T (n) =
IITM-CS5800-2017 : Advanced Data Structures and Algorithms (Section 2)
Practice Problem Sheet #3
Released On : Sep 28
Matroids and Greedy Algorithms
1. Prove that, in a matroid, the independent sets w
Module 4
1
Eigenvalues and Eigenvectors
A
Given a linear operator X
X, a scalar F is called an eigen value of A iff the equation Ax = x
has a non-zero solutions; put another way, ker(A I) := cfw_x X
Module 5
1
Sesqui/Bi-Linear forms
Sright
tr
1 Suppose X is right K-module so that X = X (
= X tr is a right K-module again. Let X X
be a right K-linear transformation. Thus to each x X we associate a
Module 4
1
Diagonalization
Now suppose in A () = ( 1 )p1 . . . ( k )pk we have pj = 1 for 1 j k (which means then there
are only linear factors in the minimal polynomial) then there are exactly k dist
Department of Mathematics, IIT Madras
MA2030
Linear Algebra & Numerical Analysis
Assignment-3
1. Check whether each of the following is an inner product on the given vector spaces.
(a) hx, yi = x1 y1
Department of Mathematics, IIT Madras
MA2030
Linear Algebra & Numerical Analysis
Assignment-1
1. Let V be a vector space over F. Show the following:
(a) For all x, y, z V, x + y = z + y implies x = z.
Bibliography
[1] J. Hefferon, Linear Algebra, http:/joshua.smcvt.edu/linearalgebra, 2014.
[2] E. Kreyszig, Advanced Engineering Mathematics, 9th Ed., John Willey & Sons, 2006.
[3] S. Kumaresan, Linear
Class Notes
MA 2030
Linear Algebra and Numerical Analysis
Arindama Singh & A. V. Jayanthan
IIT Madras
Caution
The writing style here is not rened.
It is only a class note, not a book.
You must learn c