Let V be the vector space of polynomials of degree at most 2 with real coefficients. Let
T R^3 be a linear transformation defined by





Find the matrix of T relative to the basis of V and the standard basis of R^3.

Find the quadratic polynomial which best fits the points and 3

Find the Jordan form ofA, where A is

1 0
2 1 1 0
0 1 2 1
0 2

You are given that is a characteristic polynomial of A.

Consider the predator-prey system given by
[xk [0.6 0.5 [xk
-0·16 1·2] Yk]

What is the long term behaviour of [xk
yk]

3. Find the singular value decomposition for the matrix A is

1
0 1


Solve the system of differential equations
with


where A is
4


Check whether the matrix

0 24
4 1 16
0

is diagonalisable.

5. Which of the following statements are true and which are false? Justify your answer.

For any normal operator

ker T ker T*.

A matrix has a unique generalized inverse.

Up to similarity, there is a unique matrix with characteristic polynomial and minimal polynomial

If A is an n x n matrix, then
=e^det

Product of two unitary matrices need not be unitary.