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No. of Printed Pages : 8 MTE-02 BACHELOR'S DEGREE PROGRAMME
(BDP) 07606
Term-End Examination June, 2014
ELECTIVE COURSE : MATHEMATICS MTE-02 : LINEAR ALGEBRA
Time : 2 hours Maximum Marks : 50
(Weightage 70%) Note : Question no. 7 is compulsory. Attempt any
four
questions from Q. No. 1 to Q. No. 6. Use of calculators is not
allowed.
1. (a) Show that Q[x], the set of all polynomials with rational
coefficients is a vector space over Q with respect to addition of
polynomials and multiplication by constant. Is the set of all
polynomials with integer coefficients a subspace of this vector
space ? Give reasons for your answer. 4
(b) Find the range space and the kernel of the linear
transformation : 6 T R4
--+ R4, T(xi, x2, x3, x4) = (x1 + x2 + x3 + x4, x1
+ x2, x3 + x4, 0) 2. (a) Show that if S and T are linear
transformations on a finite dimensional vector space, then rank
(ST) rank (T). Also give examples of linear transformations S and T
for which rank (ST) < rank (T). 4
MTE-02 1 P.T.O.
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(b) Find the eigenvalues and eigenvectors of the matrix
3. (a)
A=
Is A
Find {1, 1
1 1 0
1 1 0
2 2
2
diagonalisable ?
the dual basis for + x, x2 1} of the (a0 + ax + a2x2
Justify your answer.
the basis vector space
E
6
4 P3 a0, al' a = 2 1
(b) Verify the Cayley Hamilton theorem for 2 1 1
A = 1 2 1 . Hence find its inverse. 4
1 1 3
(c) Check whether the set S = {(al, a2, ..., an) E Rn I = 1 +
a2}
is a subspace of Rn or not. 2
. (a) Check whether the quadratic forms 2x2 + 3y2 + 5z2 4xz 6yz
and
4x2 + 3y2 + z2 6xy 2xz are orthogonally equivalent.
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(b) Let V = {(a, b, c) E R3 a + b = c} and
W = {(a, b, c) E R3 I a = b} be subspaces of
R3. (i) Find the dimensions of V, W and
V n w. (ii) Is R3 = V 9 W ? Justify your answer.
4
(c) Is the following matrix Hermitian ? 1 i 0
A= i 1 1i 0 1+ i 2
Is it unitary ? Justify your answer. 2
5. (a) Let V = {(a, b, c, d) R4 la+b+c+d= 0}
and W = {(a, b, c, E R4 I a = b, c = d} be
subspaces of R4. (i) Check that W is a subspace of V. (ii) Find
the dimension of VAV. (iii) Check whether (1, 1, 1, 3) + W and
(-1, 2, 0, 1) + W represent the same element of V/W. 5
(b) Find an orthonormal basis for C3 by
applying Gram Schmidt orthogonalisation process to the basis
{(1, i, 0), ( i, 0, 2), (0, i, 2)1. 5
MTE-02 3 P.T.O.
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6.
7.
(a) Find the the quadratic
9x2 and its
(b) Find the
Hence,
Which of the which are false a short proof
orthogonal canonical reduction of form
+ 7y2 + 11z2 8xy + 8x2 principal axis. 6
adjoint of the matrix 1 0 3
1 1 0
0 1 2
find its inverse. 4
following statements are true and ? Justify your answer either
with
or with a counter-example. 10
(i) If W and U are subspaces of the vector space V having the
same dimensions, then U = W.
(ii) If V is a vector space over K and f : V > K is a
non-zero linear function, then f is onto.
(iii) A 3 x 3 matrix with real entries has a real
eigenvalue.
(iv) If T is a linear transformation on an inner product space V
and T * T = 0, then T = 0.
(v) The rank and signature of a quadratic form are always
equal.
MTE-02 4
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MTE-02 5 P.T.O.
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4.
MTE-02 6
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MTE-02 7 P.T.O.
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MTE-02 8 5.000
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