MT 5508/MT 5502 – LINEAR ALGEBRA

Date : 03-05-2012 Dept. No. Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

Answer ALL questions: (10 x2 =20 marks)

If V is a vector over a field F, Show that (-a)v = a(-v)= -(av), for a F, v
Is the Union of two subspaces is a subspace?
Show that the vectors (1,0,-1), (2,1,3),(-1,0,0) and (1,0,1) are linearly dependent in .
Determine the following mapping is a vector space homomorphism: T : by T(a,b)=ab.
Define inner product space.
Define orthonormal set in an inner product space.
Prove that is orthogonal.
For A,B F_n and then prove that tr (A+B) = tr A + tr B.
Define Hermitian and skew-Hermitian.
Find the rank of the matrix over field of rational numbers.

PART – B

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

Prove that a non empty subset W of a vector space V over F is a subspace of V if and only if aw₁+bw₂ W , for all a,b F , w₁,w₂
If v₁,v₂,. . .,v_n V are linearly independent , and if v V is not in their linear span, Prove that v₁,v₂,. . .,v_n are linearly independent.
Find the coordinate vector of (2,1,-6) of R³ relative to the basis {(1,1,2),(3,-1,0),(2,0,-1)}.
Prove that T : defined by T(a,b) = (a-b, b-a,-a) for all a,b is a vector space homomorphism.
State and Prove Schwarz inequality.
If is an eigen value of T A(v), then for any polynomial f(x) F[x] , f() is an eigen value of f(T).
If A and B are Hermitian , Show that AB + BA is Hermitian and AB-BA is skew-Hermitian.
Show that the system of equations x₁+2x₂+x₃ = 11, 4x₁+6x₂+5x₃ = 8, 2x₁+2x₂+3x₃ = 19 is inconsistent.

PART – C

Answer any TWO questions: (2 x20 =40 marks)

(a) Let V be a vector space of finite dimension and let W₁ and W₂ be subspaces of V such that

V = W₁+W₂ and dim V = dim W₁+dim W₂. Then prove that V = W₁W₂. (10 + 10)

(b) If A and B are subspaces of a vector space V over F , Prove that

(a) If V is a finite dimensional inner product space and w is a sub space of V, prove that .

(b) Show that is invertible if and only if the constant term of the minimal polynomial

for T is not zero. (10 + 10)

21.(a) If are distinct eigen values of and if v₁,v₂,. . .,v_nare eigen vectors of

T belonging to respectively, then v₁,v₂,. . .,v_n are linearly independent over F.

(b) If A,B F_n, where F is the complex field, then

(i) , (ii) , , (iii) , (iv) .

(10 + 10)

(a) 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.

(b) (i) If is skew-Hermitian , Prove that all of its eigen values are pure imaginaries.

(ii) Prove that the eigen values of a unitary transformation are all of absolute value one.

(10 + 10)

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