MA2030: Linear Algebra and Numerical Analysis
Assignment-6
January May 2012
1. Consider R3 with the standard inner product. In each of the following, nd orthogonal vectors
obtained from the given vectors using Gram-Schmidt orthogonalization procedure:
(a)
Chapter 1
Vector Spaces
1.1 What is a vector space?
The notion of a vector space is an abstraction of the familiar set of vectors in two or three dimensional Euclidean space.
Let R denote the set of all real numbers. Let C denote the set of all complex nu
Classnotes - MA2030
Linear Algebra
Arindama Singh
Department of Mathematics
Indian Institute of Technology Madras
This has been prepared basing on the classnotes of A. V. Jayanthan and A. Singh
Contents
1
2
3
4
5
Vector Spaces
1.1 What is a vector space?
Theorem 1.7. Let V be a vector space. Let v1 , . . . , vn V. Then
1. cfw_v1 , . . . , vn is linearly dependent iff 1 v1 + + n vn = 0 where at least one i is nonzero.
2. cfw_v1 , . . . , vn is linearly independent iff 1 v1 + + n vn = 0 implies 1 = = n =
Chapter 3
Linear Transformations
3.1 Linear maps
Let V and W be vector spaces over F. A function T : V W is said to be a linear transformation (or a linear map) iff T (x + y) = T (x) + T (y) and T (x) = T (x) for every x, y V and
for every F.
A bijective
Chapter 2
Inner Product Spaces
2.1 Inner Products
Let V be a vector space over F, which is either R or C. We define an inner product in a vector
space V by accepting some of the fundamental properties of the scalar product in R2 or R3 .
An inner product o
LIBRARY OF
WELLESLEY COLLEGE
PRESENTED BY
TVof Horafo-rd
HIGHER ALGEBRA
A SEQUEL TO
ELEMENTARY ALGEBRA EOR SCHOOLS.
s.
HIGHER ALGEBRA
A SEQUEL TO
ELEMENTARY ALGEBRA FOR SCHOOLS
BY
H.
S.
HALL,
M.A.,
FORMERLY SCHOLAR OF CHRIST'S COLLEGE, CAMBRIDGE,
MASTER O
Module 5
1
More on Sesqui/Bi-Linear forms
s
1. Consider now a sesquiliner form X X
K and introduce s (x, u) := s(u, x). Then s is
again a sesquilinear form (s (x, u) = s(u, x) = s(u, x) = s(u, x) = s (x, u)
Let Y CenK; this means Y = Y for each K and he
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 distinct eigenvalues, with nj
linearly independent eigenvec
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 left K-linear transformation
sright (x)
X K. Recording
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 | Ax = x is a non-zero subspace
of X. Thus for dim 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) = s(u, x). Then s(x, u) + s(x, u) =
s(x, u) + s(u, x)
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 + u, x( + ) = x + x
etc; we also can, and do, write x X
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, . . . , 0] if m = n
(ii)
. . . (1)
1
D=
.
0
0
.
r
.
.
0
Department of Mathematics, IIT Madras
MA2030
Linear Algebra & Numerical Analysis
Assignment-4
1. In each of the following determine whether T : R2 R2 is a linear transformation:
(a) T (, ) = (1, )
(b) T (, ) = (, 2 )
(c) T (, ) = (sin , 0)
(d) T (, ) = (|
Classnotes - MA2030
Linear Algebra
Arindama Singh
Department of Mathematics
Indian Institute of Technology Madras
This has been prepared basing on the classnotes of A. V. Jayanthan and A. Singh
Contents
1
2
3
4
5
Vector Spaces
1.1 What is a vector space?
MA2030: Linear Algebra and Numerical Analysis
Assignment-5
January May 2012
1. Recall the notation:
Pn ([0, 1], R) is the space of all polynomial functions from [0, 1] to R.
R([0, 1], R) is the space of all Riemann integrable functions from [0, 1] to R.
C
MA2030: Linear Algebra and Numerical Analysis
Assignment-4
January May 2012
In the following V, W, V1 , V2 denote nite dimensional vector spaces over F, which is R or C.
1. Let E1 = cfw_u1 , . . . , un and E2 = cfw_v1 , . . . , vm be bases of V1 and V2
Linear Algebra and Numerical Analysis
Assignment 1
System of Linear Equations, Rank and Basics of Vector spaces
July November, 2014
2 1 3
0 2
3 2 1 1 0
.
1. Find the solution set of the system of linear equations Ax = 0, where A =
1 0 5
1 4
4 3 1 1 5
1 1
Linear Algebra and Numerical Analysis
Assignment-9
July-November, 2014
1. Consider R3 with the standard inner product. In each of the following, nd orthogonal vectors obtained
from the given vectors using Gram-Schmidt orthogonalization procedure:
(a) (1,
Linear Algebra and Numerical Analysis
Assignment-7
Some of the exercises here are written for general inner product spaces. You may either try to solve
assume that the space in consideration is a subspace of Rn for some n, or for the general case. F
denot
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 correct writing style from your teacher.
1
Vector Spaces
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 Algebra - A Geometric approach, PHI, 200.
[4] S. Lang,
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.
(b) For all , F, x V, x 6= 0, x 6= x iff 6= .
2. In ea
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 for x = (x1 , x2 ), y = (y1 , y2 ) on R2 .
R1
(b) hf, g
Department of Mathematics, IIT Madras
MA2030
Linear Algebra & Numerical Analysis
Assignment-5
1. Let V be a non-trivial real vector space. Let T : V R be a non-zero linear map. Prove or disprove:
T is onto iff null T = dim V 1.
2. Let cfw_v1 , . . . , vn
Department of Mathematics, IIT Madras
MA2030
Linear Algebra & Numerical Analysis
Assignment-2
1. Answer the following questions with justification:
(a) Is every subset of a linearly independent set linearly independent?
(b) Is every subset of a linearly d
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) X as xi X at each i I. For x, u X I , F, define
x + u X