083/2023

Maximum : 100 marks

Time : 1 hour and 30 minutes

1. Let V=R’ be a vector space over R with usual addition and scalar multiplication. Then
which of the following is not a subspace of V?

(^) W, ={(X,, 2५, 5५) : 21 + ४५ ८0, 1 + 2५ - 328 = 0}
(B) 155 5 (७ ,29)%3) : 1 = 1, 2 = 1}
(© We ನಯ ಖು. ५७): ५ + ಖ್ಯ 5 0)

(D) Wy = {(%,% 2,23) #22 = 8a,%3 = Bay}

2. Let (e,e,..,¢e,} 18 a standard basis of நூ, Then ({f,f,..,f,} where
- : ष if:
15911 Gy, fy = My Cy + Ugg Coser fy = Uy Cy + Ayn Cy +++. +A, Cy 18 a basis of R” if and only if :

൧ സൃ ന... = 0 (छ) ०। *०५५...०,. #0

Any mn

0 6 بريه + ... + ووه + ررك ‎(D)‏ 0 > نه + ...+ ووه + ‎yy‏ ©)

8. The linear transformation ற்‌ defined by 41 (1, 0, 0) = (1, 2, 0, 4) ,
T (0, 1, 0) = (2, 0, -1, -3), TO, 0, 1) = (0, 0, 0, 0). Then which of the following is true?
(4) 1 (01, 2, 08) = ("1 + 20९, 15 095 431 +392)
(8) 1 (01, 39, 39) 5 (01 + 239, 20117 095 401 7339)
(@ 1 (01102, 28) = (21 - 202, - 00, Yar 494 +399)
त) 101, 2 › 29) = (0 - 92) 21, - 92, - 401 - 80)

4. Let T:R°—R’* be a linear transformation defined by 7 (X1,%_.%3) = (1, 8, 0). If r =
dimension of range space of T and n = dimension of null space of T, then the value of 2” +3”
is:

(A) 5 (8) 7
(0 11 0) 18

5. Let Abe an eigen value of a non-singular matrix A. Then eigen value of adjoint of A, adj A is:

1
5 ® لما بم
4إ 0
© رو 0 0(
‎A 3‏

[P.T.0.]