B:-nonhomogeneous and inconsistent
C:-consistant and has unique solution
D:-consistant and has infinite solution
Correct Answer:- Option-D
Question75:-Which of the following map ‘T:RR*(3)->RR° is a linear functional ?
A T(x,y,2)=5°
B:-T(x, y, 2) = x*(2)°
C:T(x,y,z)=-2x+y"
D:-T(x,y, z)=xy+6°
Correct Answer:- Option-C
Question76:-Let V be a vector of dimension 15 over a field F and W be a subspace of V. If dim W = 3, then ‘dim(V/MW)=*
‏3ھ‎
‎B:-5
‎C:-8
‎D:-12
‎Correct Answer:- Option-D
Question77:-An ‘nxxn’ matrix is diagonalizable if
A:-all the eigen values are real and distinct
B:-all the eigen values are real and non-negative
C:-all the eigen values are real and non-zero
D:-all the eigen values are non-zero rational numbers
Correct Answer:- Option-A
Question78:-Let ‘1<=pA:-Minkowski's inequality
B:-Jensen's inequality
C:-Cauchy's inequality
D:-Bessel's inequality
Correct Answer:- Option-B
Question79:-Which of the following is false ?
A:-I*(p)’_ is a Hilbert space, where *1<=p<=o0*
B:-Closed subspace of a Hilbert space is a Hilbert space
C:-The quotient of a Hilbert space by one of its closed subspace is again a Hilbert space
D:-A complete normed space with its norm satisfies the parallelogram law is a Hilbert space
Correct Answer:- Option-A
Question80:-Let A*in’ BL(H) and A* be the adjoint of A. A is unitary if

A:-AA* = A*A
B:-AA* *!=* A*A
C:-A* = 87(-1)
D:-A*=A

Correct Answer:- Option-C
Question81:-Which of the following is not a Banach space ?
A:-Finite dimensional normed spaces

8:-17(9) with norm ` || `||_(0)', where *1<= 0<=00 `
:- ० (00)
‏'(ع)(م) ا ٭-:5‎ ۷٣۲ ‏معمم عط‎ '||' `||_(0)`, where E is a Lebesgue measurable subset of ‘RR’ and *1<=p<=o0*

Correct Answer:- Option-C
Question82:-Let X be an inner product space and let x, y ‘in’ X. Then the parallelogram law is
[xt yl] (2) + ||| ^ (2) = 2 (||| ^ (2) + ||| ^ )2((“
[+| ^ (2) + ||-/||^ (2) = ||| ^ (2) + ||#||^ (2)
॥+/॥॥<5||+2||+||[2+/॥

0:-` 54|| ^ (2) + ||| ^ (2)) <= |||

Correct Answer:- Option-A
Question83:-Let X and Y be normed spaces and F : X ‘->* Y be a linear map. The following conditions are equivalent except
one. Which of the following is not equivalent to others ?

A--F is continuous at 0

B:-F is continuous on X

C:-‘al|x||<=||F(x)||*, for all x‘in’X and some a > 0

D:-F is uniformly continuous on X

Correct Answer:- Option-C