**[OBJECTIVE]**

**Subject:** Linear Algebra

**Time Allowed: **10 Minutes

**Maximum Marks**: 10

**NOTE:** Attempt this Paper on this Question Sheet only. Please encircle the correct option. Division of marks is given in front of each question. This Paper will be collected back after expiry of time limit mentioned above.

__Part-I Encircle the correct option, cutting and overwriting is not allowed. (10)__

Every non-zero vector in a vector space over a field is …

a) Linearly independent b) Linearly dependent

c) Basis d) Subspace

A complete Normed space is called…

a) Normed space b) Bancach space

c) Hilbert space d) Not Given

The number of subgroups of the cyclic group of order 49 is.

a) 1 b) 2

c) 3 d) Not Given

The number of generators in a cyclic group of order 15 is.

(a) 8 (b) 10

(c) 3 (d) 12

e) Not Given

The order of the smallest Abelian group is.

a) 3 b) 2

c) 1 d) Not Given

Let T: R^{15} —» R be a linear transformation. If the dimension of the null space of 7 is 12, then the dimension of R(T) is

a) 15 b) 5

c) 3 d) Not Given

In the group (Z, o) of all integers, where aob = 1/2ab for a, b € Z, the inverse of 4 is

a) 0 b)

c) 3 d) 4

e) Not given

The group C_{2} X C_{2} is isomorphic to.

a) C_{4} b) V_{4}

c) S_{3} d) Not given

The dimension of the vector space of reals over complex numbers, namely &(C) is.

a) 1 b) 2

c) infinite d) Not Given

**[SUBJECTIVE]**

**Subject:** Linear Algebra

**Time Allowed: **2 Hours 45 Minutes

**Maximum Marks**: 50

**NOTE:** ATTEMPT THIS (SUBJECTIVE) ON THE SEPARATE ANSWER SHEET PROVIDED.

__Part-II Give short details of each of them, each answer carries equal marks. (20)__

**Q#**1: Prove that every finite dimensional vector space contains a basis.

**Q#**2: Let G =< a, b| a^{3} = b^{2} = (ab)^{ 2} =e >. Find all subgroups of G.

**Q#**3: Define reducible and irreducible representations and state Schurs Lemma without proof.

**Q#**4: Prove that Q(√2) = {a + √2b : a, b e ∈ R} is a vector space over R, where R is the set of real numbers.

**Q#**5: State and explain Gram-Schmidt method with one example.

__Part-III Give brief answers, each answer carries equal marks. (30)__

**Q#**1: Solve the following system of equations

2x – 2y + 3z = 8

x + 6y + 3z = -3

2x + 6y + 12z = -3

**Q#**2: Prove that in an n-dimensional vector space, any set of n+1 or more vectors is linearly dependent.

**Q#**3: For a system of linear equations in n unknowns, if augmented matrix M = [A, B]. then prove that:

**(a)** The system has a solution if and only if rank (A) = rank (M).

**(b)** The solution is unique if and only if rank (A) + rank (M) = n.

**Q#**4: Diagonalize the given matrix (if possible).

**Q#**5: Prove that the set of all bijective mappings on a set G = (1,2,3} to G forma group.