Annamalai University 2008-3rd Year B.Sc Mathematics " 710 VECTOR CALCULUS AND LINEAR ALGEBRA " ( - IV ) ( PART - III - A - MAIN ) ( ) 5236 - Question Paper

4

If
		-3 -2 2 '
	A =	6 5-2
		- -6 -2 5 -
find A-1.
Find the rank of the matrix
	1 3	4 5 6	7
	4	5 6 7	8
	5	6 7 8	9
	10	7 8 9	10
	7	8 9 1	2

(b) If

cos x -sin x 0 sin x cos x 0

0 0 1

prove that

f(x + y) = f(x) f(y).

Name of the Candidate :

5 2 3 6 B.Sc. DEGREE EXAMINATION, 2008

(MATHEMATICS)

(THIRD YEAR)

(PART - III - A - MAIN)

(PAPER - IV)

710. VECTOR CALCULUS AND LINEAR ALGEBRA

(Including Lateral Entry )

December ]    [ Time : 3 Hours

Maximum : 100 Marks

Answer any FIVE questions.

All questions carry equal marks.

(5 x 20 = 100)

f dr

where

f = (sin y) i + x(l+cosy)j and c is the circle

in the xy - plane.

(b) Prove that

(i) A( f(r) ) = [f'(r)]7/r and (ii) A(rⁿ) = nrⁿ² r.

2. (a) Show that (rⁿ r) is solenoidal if n = -3. (b) Prove that

curl ((f) F) = (grad (f>) x F + <f> (curl F).

f (x² - y²)i + 2xy j over the box bounded by the planes

x = 0, x = a, y = 0, y = b, z = 0, z = c if the face z = c is cut.

4.    (a) Prove that

1 ab c (a + b)

1 be a (b + c)

1 ca b (c + a)

(b) Using determinants, solve the equations x + y + 2z = 4 2x - y + 3z = 9 3x - y - z = 2.

5.    (a) Prove that any Hermitian matrix can be

uniquely expressed as A + iB where A is a real symmetric and B is real skew symmetric.

9.    (a) Prove that any two bases of a finite

dimensional vector space V have the same number of elements.

(b) Prove that any vector space of dimension n over a field F is isomorphic to

V (F) = Fⁿ.

n^{v 7}

10.    (a) Let V be a finite dimensional vector space

over a field F. Let W be a subspace of V. Then, prove that

(i) dim W < dim V

and (ii) dim |-| = dim V - dim W.

(b) Let V be a vector space over F. Let

S = {V_r V₂, ..., V_n} c V

if and only if, S is a maximal linearly independent set.

7.    (a) Find the eigen value and the corresponding

eigen vectors of the matrix

6 -2 2--2 3 -1

- 2 -1 3-

(b) Test for consistency the following system of equations. If it is consistent, find the solution:

x 4y 3z = -16

4x - y + 6z = 16

2x + 7y + 12z = 48

5x - 5y + 3z = 0.

8.    (a) Show that

{ (1, 1, 0), (1, 0, 1), (0, 1, 1), }

a

is a basis for R over R.

(b) If A and B are subspaces of a vector space V, prove that A + B and A n B are subspaces of V.

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