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 R² 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 R³ 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 R³ 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 W₁ and W₂

if and only if V = W₁ + W₂ and W₁ W₂ = {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 R⁴ 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)

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Loyola College B.Sc. Mathematics Nov 2012 Linear Algebra Question Paper PDF Download