MT 5508 / 5502 – LINEAR ALGEBRA

Date & Time: 06/05/2009 / 1:00 – 4:00 Dept. No. Max. : 100 Marks

SECTION – A

Answer ALL the questions. (10 X 2 = 20)

Let V be a vector space. Prove that (– a)v = a(– v) = – (av) for any a Î F and vÎ V.
Prove that any non-empty subset of linearly independent vectors is linearly independent.
Show that the vectors (1, 0, -1), (2, 1, 3), (–1, 0, 0) and (1, 0, 2) are linearly dependent in R³.
Verify that the map T : R² ® R² defined by T(x, y) = (2x + y, y) is a homomorphism.
If V is an inner product space, then prove that
Define eigenvalue and eigenvector of a linear transformation.
If l Î F is an eigenvalue of T Î A(V), prove that l is a root of the minimal polynomial of T over F.
For A, B Î F_n, prove that tr (A + B) = tr (A) + tr (B).
If A and B are Hermitian, show that AB + BA is Hermitian.
Prove that the eigenvalues of a unitary transformation of T are all of absolute value 1.

SECTION – B

Answer any FIVE questions only. (5 X 8 = 40)

Prove that the union of two subspaces of a vector space V over F is a subspace of V if and only if one is contained in the other.
If V is a vector space of finite dimension and is the direct sum of its subspaces U and W, then prove that dim V = dim U + dim W.
Let U and V be vector spaces over a field F, and suppose that U has finite dimension n. Let {u₁, u₂, . . . u_n} be a basis of U and let v₁, v₂, . . . v_n be arbitrary vectors in V. Prove that there exists a unique homomorphism T: U ® V such that T(u₁) = v₁, T(u₂) = v₂, . . . , T(u_n) = v_n.
Define an orthonormal set. If {w₁, w₂, . . . w_n} is an orthonormal set in an inner product space V, prove that .
Prove that T Î A(V) is invertible if and only if the constant term of the minimal polynomial for T is not zero.

Let V = R³ and suppose that is the matrix of T Î A(V) relative to the standard basis v₁ = (1, 0, 0), v₂ = (0, 1, 0), v₃ = (0, 0, 1). Find the matrix of T relative to the basis w₁ = (1, 1, 0), w₂ = (1, 2, 0) and w₃ = (1, 2, 1).
Show that any square matrix can be expressed uniquely as the sum of a symmetric matrix and a skew-symmetric matrix.
Investigate for what values of l, m, the system of equations x₁ + x₂ + x₃ = 6,
x₁ + 2x₂ + 3x₃ = 10, x₁ + 2x₂ + l x₃ = m over a rational field has (a) no solution (b) a unique solution (c) an infinite number of solutions.

SECTION – C

Answer any Two questions. (2 X 20 = 40)

(a) Let T : U ® V be a homomorphism of two vector spaces over F and suppose that U has finite dimension. Then prove that

Dim U = nullity T + rank of T

(b) If W₁ and W₂ are subspaces of a finite dimensional vector space V, prove that dim (W₁ + W₂) = dim W₁ + dim W₂ – dim (W₁ Ç W₂)

(a) Prove that T Î A(V) is singular if and only if there exists an element v ¹ 0 in V such that T(v) = 0.

(b) Prove that every finite-dimensional inner product space V has an orthonormal set as a basis. (8+12)

(a) If T be the linear transformation on R³ defined by
T(a₁, a₂, a₃) = ( a₁ + a₂ + a₃, – a₁ – a₂ – 4a₃, 2a₁ – a₃), find the matrix of T relative to the basis v₁ = (1, 1, 1), v₂ = (0, 1, 1), v₃ = (1, 0, 1).

(b) 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 Ç W₂ = (0). (8+12)

(i) (S+T)^* = S^* + T^* (ii) (ST)^* = T^*S^* (iii) (iv) (T^*)^* = T

(b) Prove that the linear transformation T on V is unitary if and only if it takes an orthonormal basis of V onto an orthonormal basis of V.

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