Find the orthogonal canonical reduction of the quadratic form  and ts principal axes. Also, find the rank and signature of the quadratic form.

Apply the Gram-Schmidt diagonalisation process to find an orthonormal basis for the subspace of C4 generated by the vectors

Let  represent the coordinates with respect to the bases B₁ = {(1,0,0), (0,1,0), (0,0,1)}, B₂ = {(1,0,0), (0,1,2), (0,2,1)}. If  find the representation of Q in terms of

Are there values of a ∈ C for which the matrix

is unitary? Justify your answer.

Find the values of a, b ∈ C for which the matrix

is Hermitian.

Using the fact that det(AB) = det(A) det(B) for any two matrices A and B, prove the identity

Find the inverse of the matrix B in part a) by using Cayley-Hamilton theorem.

For the following matrices, check whether there exists an invertible matrix P such that P^-1AP is diagonal. When such a P exists, find P.

Find the solutions to the following system of equations by reducing the corresponding augmented matrix to row-reduced echelon form. 

2a+3b+4c+d=8

a+2b+2c+2d=3

a-b+c+3d = 3

Show that, if A is any n x n matrix with real entries, then there is a n x n symmetric matrix S and a n x n skew symmetric matrix S' such that A = S + S'. 

Is the matrix of the linear operator T non-singular? Justify your answer.

Write down the matrix of T on W w.r.t the basis S.

Check that T(W)⊂ W.

Let W = [S] and let T: V → V be the function defined by

Check that T is a linear transformation on V.

Check that S is a linearly independent set over R. (Hint: Consider the equation

Let V be the set of all functions that are twice differentiable in R and

S = {cosx, sinx, xcosx,xsinx}.

Find the direction cosines of the perpendicular from the origin to the plane 

Check that (fog)(x) = x for x ∈ R \{2} and (gof) (x) = x for x ∈ R \{-1

Check that g: R\{2} → R given by g(x) = is well defined and 1 - 1. Further, check that g(x) -1 for any x ∈ R.

Check that f(x) ≠ 2 for any x ∈ R.