LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

**B.Sc.** DEGREE EXAMINATION – **MATHEMATICS**

FIFTH SEMESTER – NOVEMBER 2012

# MT 5508/MT 5502 – LINEAR ALGEBRA

Date : 08/11/2012 Dept. No. Max. : 100 Marks

Time : 9:00 – 12:00

__PART – A__

** **

**Answer ALL questions: (10 x 2 = 20 marks)**

** **

- Define a vector space over a field F.
- Show that the vectors (1,1) and (-3, 2) in R
^{2}are linearly independent over R^{, }the field of real numbers. - Define homomorphism of a vector space into itself.
- Define rank and nullity of a vector space homomorphism T: u®
- Define an orthonormal set.
- Normalise in R
^{3}relative to the standard inner product. - Define a skew symmetric matrix and give an example.
- Show that is orthogonal.
- Show that is unitary.
- Define unitary linear transformation.

__PART – B__

**Answer any FIVE questions: (5 x 8 =40 marks)**

- Prove that the intersection of two subspaces of a vector space v is a subspace of V.
- If S and T are subsets of a vector space V over F, then prove that
- S T implies that L(S) ≤ L(T)
- L(L(S)) = L(S)
- L(S U T) = L(S) + L(T).
- Determine whether the vectors (1,3,2), (1, -7, -8) and (2, 1, -1) in R
^{3}are linearly dependent on independent over R. - If V is a vector space of finite dimension and W is a subspace of V, then prove that

dim V/W = dim V – dim W.

- For any two vectors u, v in V, Prove that .
- If and l Î F, then prove that l is an eigen value of T it and only if [l I – T] is singular.
- Show that any square matrix can be expressed as the sum of a symmetric matrix and a skew symmetric matrix.
- For what values of T, the system of equations over the rational field is consistent?

__PART – C__

** **

**Answer any TWO questions: (2 x 20 = 40 marks)**

- a) Prove that the vector space V over F is a direct sum of two of its subspaces W
_{1}and W_{2}

if and only if V = W_{1} + W_{2} and W_{1} W_{2} = {0}.

- b) If V is a vector space of finite dimension and is the direct sum of its subspaces U and

W, than prove that dim V = dim U + dim W. (10 + 10)

- If U and V are vector spaces of dimension m and n respectively over F, then prove that the

vector space Hom (U, V) is of dimension mn.

- Apply the Gram – Schmidt orthonormalization process to obtain an orthonormal basis for

the subspace of R^{4} generated by the vectors (1, 1, 0, 1) , (1, -2, 0, 0) and (1, 0, -1, 2).

- a) Prove that TÎA(V) is singular if and only it there exists an element v ≠ 0 in V such that

T(v) = 0.

- b) Prove that the linear transformation T on V is unitary of and only if it takes an

orthonormal basis of V onto an orthonormal basis of V. (10 +10)

** **

**Latest Govt Job & Exam Updates:**