3582425736

T

(6

: a lal

nd

«)

:ar

ito

ts)

«s)

«)

<S)

cn

USN

Time: 3 hrs.

08EC046

M.Tcch. Degrec Examination, Dec.08/Jan.09 Linear Algebra

Notę: Answer any FTVEfuli questions

] a. Solve the following system of cquations: x + 2y-3z=I 2x+5y-8z = 4

3x + 8y-13z = 7 by Gauss elimination method. b. Reduce the following matrix:

12—31    2

2    4    -4    6    10    to Rowreduccd Echclon form.

3    6    -ó    9    13

2 1 4    5

-4    1

6    -3

2 a. Express M as a linear eombination of the matrices A. B, C where

c. Find the LU factorization with /.. = 1 for the mairix A =

Max. Marks: 100

(06 Marks)

(06 Marks)

1 -1 4 -1 4 4 3 1

(OS Marks)

k

ts)

«)

	4 7"		1 1		1 2		i i
M =	7 9	, A =	1 1	,B =	— n ro _i	, c=	i l_

(06 Marks)

«)

or

if

«)

V.

cd

ts)

C.

3 a.

b.

c.

<s)

4 a.

«)

Prove that the set W = {(x,y,z)/x-3y + 4z=0} of the vector space V₃(R)is a subspace of

V₃(R).    (07    Marks)

Prove that the inversc of two    subspaces    of a    vcctor    space V is a subspace of V. Is it true in

the case of union of two subspaces?    Justify your    answer.    (07    Marks)

If a, p, y are lincarly independent in V(F), prove that the vectors a+p, a-P, a-2p+y are also linearly independent.    (06    Marks)

Prove that any two bases of a finite dimensional vector space V have the same nuniber of elements.    (06    Marks)

Lct T:U -> V be a linear map. Then prove that

i)    R(T) is a subspace ofV.

ii)    N(T) is a subspace of U.

iii) T is 1 — 1 iff the nuli space (N(T)) is a zero subspace.    (08 Marks)

Prove that T:U -> V of a vector space U to a vector space V over the same field F is a linear transformalion if and only if V u, p 6 U and C₍, cF

«)

T(C_]a + C₂P) = C_IT(a) + C₂T(p)

Find the eigen space of the linear transformation

T : R^J -> R³ defined by

T(x, y, z) - (2x-i-y, y-z, 2y+4z)

Find the linear transformation relative to the bases,

(06 Marks)

(07 Marks)

B, ={(!,!),(-1,1)}, B₂ = {(!, 1,1),(1,-1,1X0,0,1)} given the matrix A_y =

1 of 2

1 2 0 1

-1 3

(07 Marks)