Download MTE-02-2008

MTE-02
Assignment Booklet
Bachelor’s Degree Programme
Linear Algebra
School of Sciences
Indira Gandhi National Open University
New Delhi
2008
Dear Student,
Please read the section on assignments in the Programme Guide for Elective Courses that we sent you
after your enrolment. A weightage of 30 per cent, as you are aware, has been earmarked for continuous
evaluation. For this you need to answer one tutor-marked assignment for this course, which is in this
booklet.
Instructions for Formating Your Assignments
Before attempting the assignment please read the following instructions carefully.
1) On top of the first page of your answer sheet, please write the details exactly in the following format :
ROLL NO:………………………….
NAME:………………………….
ADDRESS:………………………….
…………………………..
…………………………..
…………………………..
COURSE CODE:………………………….
COURSE TITLE:………………………….
ASSIGNMENT NO:………………………
STUDY CENTRE:………………………...
DATE:…………………….………………
PLEASE FOLLOW THE FORMAT ABOVE STRICTLY TO FACILITATE EVALUATION
AND TO AVOID DELAY.
2) Use only foolscap size writing paper (but not of very thin variety) for writing your answers.
3) Leave a 4 cm. margin on the left, top and bottom of your answer sheet.
4) Your answers should be precise.
5) While solving problems, clearly indicate which part of which question is being solved.
6) The assignment responses are to be submitted to your Study Centre Coordinator by November, 2008,
not later.
Please keep a copy of your answer sheets.
Wish you luck!
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ASSIGNMENT
(To be done after studying the course material.)
Course Code : MTE-02
Assignment Code : MTE-02/TMA/2008
Total Marks : 100
1. Which of the following statements are true? Give reasons for your answers.
i)
A matrix A is diagonalisable if A I  0 for some R .
ii)
The geometric representation of {2, 4}  [0, 1] is a line segment.
iii)
R is a vector space over C with respect to the usual operations of addition and multiplication.
iv)
There is one and only one inner product defined from R5  R5 to R .
v)
For any two linear transformations S and T from R n to R n , every eigenvector of ST is an
eigenvector of TS.
vi)
For a 2  2 matrix A, A = Adj (Adj A).
vii)
For any linear transformation T : R[x]  Rn, dim (Ker T) + dim R(T) = n.
viii) If A is an invertible n  n matrix, then (AB) (B) , for any n  m matrix B.
ix)
Every unitary matrix is Hermitian.
x)
If the signature of
n
a x
i
2
i
is positive, then ai > 0  i 1,
(20)
, n.
i 1
2. a)
Check whether the operation  , defined by a  b = (ab – a)b, is a binary operation on R.
Further, is it commutative? Is it associative? Give reasons for your answers.
(3)
b)
Find the vector equation of the line passing through 2i – 3j and i + j + k. Which point on this
1 1 1
,
,
line has direction cosines
, and why?
(3)
3 3 3
c)
Let U, V, W be finite-dimensional subspaces of a real vector space. Show that
d)
e)
3. a)
dim U + dim V + dim W – dim (U + V + W)  max {dim (U  V), dim (V  W), dim (U  W)}.
(4)
How many 3  4 matrices over R are there whose columns are linearly independent? Give
reasons for your answer.
(2)
1 0
0 1
Find a basis for the row space of the matrix 
 3 0

 5 1
0
0
2
0
0
0
0
0
1
2 
.
0

0
(3)
Let A  Mn (C), and let C(A) {BM n (C) AB  BA} be the commutant of A. Show that
C(A) is a subspace of Mn (C).
(2)
3
b)
Let A  Mn (R). Check whether R(At A) = R(At) or not.
c)
Let V be a vector space of dimension n. Let S and T be two linear transformations from V to V
such that S2 = T2 = 0 and ST + TS = I. Prove that
i)
Ker S = S(Ker T), Ker T = T(Ker S) and V = Ker S + Ker T;
ii)
n is even;
iii)
d)
(4)
0 1 
if n = 2, then V has a basis with respect to which S and T are represented by 
 and
0 0 
0 0 
(10)
1 0  , respectively.


Obtain the dual basis of the basis {5, (5 – x), x2, (x – 3)3} of P3(R), the vector space of all
polynomials in x of degree at most 3.
(4)
1.0001
1 
 1

Prove that 1.0001
1
1.0001 has one positive and one negative eigenvalue.
 1
1.0001
1 
(3)
b)
Let A and B be n  n unitary matrices. Show that det  A  B  2n .
(4)
c)
If A  Mn (C) such that Ar = I for some r > 0, then show that A is diagonalisable.
(3)
2 2 1
Check whether A = 1 3 1  is diagonalisable. If it is, diagonalise it. If A is not
1 2 2 
diagonalisable, find Adj (A).
(6)
4. a)
d)
e)
Can Cramer’s rule be used for solving the following system of equations? If yes, apply it to get
the solution. Otherwise obtain the solution by Gaussian elimination.
ax  by  cz  0, x  y  z  0,(b  c) x  (a  c)y  (a  b)z  0, where a, b, c are distinct real
numbers.
5. a)
(4)
Let p and q be polynomials of degree n in R[x]. Define
<p, q> = p(x0) q(x0) + p(x1) q(x1) +
numbers.
+ p(xn) q(xn), where x0,
, xn are distinct fixed real
Check whether or not < , > defines an inner product on Pn(R).
(5)
b)
Reduce the quadratic form Q  3x 2  2y2  z 2  4xy  4yz to its normal canonical form. What is
its rank and signature?
(5)
c)
Reduce x 2  8xy  5y 2 16 to standard form. Hence identify the conic it represents. Also,
roughly trace the conic represented by the equation given above.
d)
Prove the following for real n  n matrices:
i)
If A is an orthogonal matrix whose eigenvalues are all different from 1, then I + A is
non-singular, and S = (I – A) (I + A)1 is skew-symmetric.
4
(6)
ii)
e)
If B is a skew-symmetric matrix, then C = (I – B) (I + B)1 is an orthogonal matrix with
no eigenvalue equal to 1.
(6)
a b 
If 
 is the matrix of a Hermitian operator, what properties must a, b, c, d satisfy? How
c d 
many such matrices are there?
(3)
5