QUESTION 1 [6]
1.1. If the nullity of the linear transformation T: Pn Mmn is 3, then determine the rank of T. [3]
1.2. Prove that a square matrix A is invertible if and only if Ois not an eigenvalue of A.
[3]
QUESTION 2 [16]
Determine whether each of the following mappings is linear or not.
= 2.1. T: 'F 'F defined by T(f) (f(x)) 2, where 'Fis the vector space of functions on IRL
[5]
= 2.2. T: Mnn Mnn defined by T(A) AC - CA, where C is a fixed n X n matrix.
[11]
QUESTION 3 [11]
T[~]m t-;, m- T[:l T[J LetT: lll.2 lll.3 defined by
= and 1] =
Find and use itto determine
QUESTION 4 [8]
Let 'F be the vector space of functions with basis S = {sint, cost, e- 2t}, and let D: 'F 'F be the
= differential operator defined by D(f(t)) f' (t). Determine the matrix [DJs representing Din the
basis S.
QUESTION 5 [11]
Find the basis and the dimension of the image of L.
QUESTION 6 [11]
Consider the bases B = {1 + x + x 2, x + x2, x2 } and C = {1, x, x2 } of P2 .
6.1. Find the change of basis matrix Ps--c from C to B.
[8]
6.2. Use the result in part (6.1) to compute [p(x)Js where p(x) = 2 + x - 3x2 .
[3]
1