1994

1) Show that f1(t)=1,f2(t)=t−2,f3(t)=(t−2)2 form a basis of P2, the space of polynomials with degree ≤2. Express 3t2−5t+4 as a linear combination of f1,f2,f3.

[10M]

2) If T:V4(R)→V3(R) is a linear transformation defined by T(a,b,c,d)=(a−b+c+d,a+2c−d,a+b+3c−3d)

For a, b, d∈R, then verify that Rank T+ Nullity T=dimV4(R).

[10M]

3) If T is an operator on R3 whose basis is B={(1,0,0),(0,1,0),(0,0,1)} such that
[T:B]=[01110−1−1−10] find the matrix of T with respect to a basis B1={(0,1,−1),(1,−1,1),(−1,1,0)}.

[10M]

4) If A=[aij] is an n×n matrix such that aii=n,aij=r if i≠j, show that [A−(n−r)H][A−(n−r+nr)I]=0
Where bii=1,bij=ρ when i≠j and ρ≠1,ρ≠11−n.

[10M]

5) Prove that the eigen vectors corresponding to distinct eigen values of a square matrix are linearly independent.

[10M]

6) Determine the eigen values and eigen vectors of the matrix A=[314026005]

[10M]

7) Show that a matrix congruent to a skew-symmetric matrix is skew-symmetric. Use the result to prove that the determinant of skew-symmetric matrix of even order is the square of a rational function of its elements.

[10M]

8) Find the rank of the matrix [0c−ba′−c0ab′b−a0c′−a′−b′−c′0] where aa′+bb′+cc′=0 and a,b,c are all positive integers.

[10M]

9) Reduce the following symmetric matrix to a diagonal form and interpret the result in terms of quadratic forms: A=[32−1223−131]

[10M]

1993

1) Show that the set S={(1,0,0),(1,1,0),(1,1,1),(0,1,0)} spans the vector space R3(R) but it is not a basis. set.

[10M]

2) Define rank and nullity of a linear transformation T. If V be a finite dimensional vector space and T a linear operator on v such that rank T2= rank T, then prove that the null space of T= the null space of T2 and the intersection of the range space and null space to T is the zero subspace of V.

[10M]

3) If the matrix of a linear operator T on R2 relative to the standard basis {(1,0),(0,1)} is (1111), what is the matrix of T relative to the basis B={(1,1),(1,−1)}?

[10M]

4) Prove that the inverse of (AOBC) is

where A, C are non-singular matrices and hence find the inverse of

[10M]

5) If A be an orthogonal matrix with the property that- 1 is not an eigen value, then show that a is expressible as ( f−S)(S+S)−1 for some suitable skew-symmetric matrix S.

[10M]

TBC

6) Show that any iwo eigen vectors corresponding to two distinct eigen values of
(i) Hermitian matrix
(ii) Unitary matrix are orthogonal.

[10M]

7) A matrix B or order n×n is of the form λ A where λ is a scalar and A has unit elements every where except in the diagonal which has elements μ. Find λ and μ so that B may be orthogonal.

[10M]

8) Find the rank of the matrix A=(1−13613−3−453311) by reducing it to canonical form.

[10M]

9) Determine the following form as definite, semi-definite or indefinite:

[10M]

1992

1) Let V and U be vector spaces over the field K and let V be of finite dimension Let T:V→ U be a linear Map. Prove that dimV=dimR(T)+dimN(T).

[10M]

2) Let S={(x,y,z)/x+y+z=0},x,y,z being real. Prove that S is a subspace of R3. Find a basis of S.

[10M]

3) Verify which of the following are linear transformations:
(i) T:B→B2 defined by T(x)=(2x,−x)
(ii) T:R_2→R3 defined by T(x,y)=(xy,y,x)
(iii) T:B2→R3 defined by T(x,y)=(x+y,y,x)
(iv) T:R→B2 defined by T(x)=(1,−1)

[10M]

TBC

4) Let T:M21→M23 be a linear transformation defined by (with usual notations) T(10)=(213415),T(11)=(610002).

Find T(xy)

[10M]

5) For what values of n do the following equations
x+y+z≤1x+2y+4z=nx+4y+10z=n2

have a solution? Solve them completely in each case.

[10M]

6) Prove that a necessary and sufficient condition of a real quadratic form X′AX to be positive definite is that the leading principal minors of A are all positive.

[10M]

7) State Cayley-Hamilton theorem and use it to calculate the inverse of the matrix →[2143]

[10M]

8) Transform the following to the diagonal forms and give the transformation employed:

[10M]

9) Prove that the characteristic roots of a Hermitian matrix are all real and a characteristic root of a skew-Hermitian matrix is either zero or a pure imaginary number.

[10M]

1991

1) Let V(R) be the real vector space of all 2×3 matrices with real entries. Find a basis for V(R). What is the dimension of V(R)?

2) Let C be the field of complex numbers and let T be the function from C3 into C3 defined by T(x1,x2,x3)=(x1−x2+2x3,2x1+x27−x1−2x2+2x3).
(i) Verify that T is a linear transformation.
(ii) If (a,b,c) is a vector in C3, what are the conditions on a,b and c so that the vector be in the range of T? what is the rank of T? (iii) What are the conditions on a, b and c so that (a,b,c) is in the null space of T? what is the nullity of T?

3) If A=(12−13), express A6−4A5+8A4−12A3+14A2 as a linear polynomial in A.

4) Let T be the linear transformation from R2 to itself defined by T(x1,x2)=(−x2,x1).
(i) What is the matrix of T in the standard ordered basis for R2?
(ii) What is the matrix of T in the ordered basis β={α1,α2} where α1=(1,2) and α2=(1,−1)?

5) Determine a non-singular matrix P such that PtAP is a diagonal matrix, where A=(012103230).

Is the matrix A congruent to a diagonal matrix? Justify.

6) Reduce the matrix [1000−110011101111][134−5−2−5−10165933−684730−78]

to echelon form by elementary row operations.

7) U is an n-rowed unitary matrix such that |I−U|≠0. Show that the matrix H defined by setting iH =(I+U) (I−U)−1 is hermitian. Also show that if eit1,eit2,………eitn be the eigen values of U, the eigen values of H are cot(α1/2),cot(α2/2),……… cot(αn/2).

8) Let A be an n-rowed square matrix with distinct eigenvalues. Show that if A is non-singular, then there exists 2n matrices X such that X2=A. What happens in case A is a singular matrix?

1989

1) Find a basis for the null space of the matrix. A=[31−1012].

2) If W is a subspace of a finite dimensional space V, then prove that dimV/W=dimV−dimW

3) Show that all vectors (x1,x2,x3,x4) in the vector space V4(R) which obey x4−x3=x2−x1 form a subspace V. Show that further that V is spanned by (1,0,0,−1), (0,1,0,1), (0,0,1,1).

4) Let P be a real skew symmetric matrix and I, the corresponding unit matrix. Show that the matrix I−P is non-singular. Also show that the matrix Q= (I+P)(I−P)−1 is orthogonal.

5) Show that an n×n matrix A is similar to a diagonal matrix if and only if the set of characteristic vectors of A includes a set of n linearly independent vectors.

6) Let r1 and r2 be distinct characteristic roots of a matrix A, and let ξi be a characteristic vector of A corresponding to ri(i=1,2). If A is Hermitian, then show that ξ1ξ2=0

7) Show that [cosθ−sinθsinθcosθ]=[1−tanθ/2tanθ/21][1tanθ/2−tanθ/21]−1

8) Verify cayley - Hamilton theorem for the matrix A=[01−23]