Test paper 1
Time allowed : 3 hours
(Allocate 20 marks for each question)
Let V be a finite-dimensional vector space. Prove that if / € £(V, V)
then
(a) dim V = dim Im / + dim Ker / ;
(b) the properties
(i) / is surjective,
(ii) / is injective,
are equivalent;
(c) V = Imfe Ker/ if and only if I m / = I m / 2 ;
(d) Im / = Ker / if and only if the following properties are satisfied
(i) f2 = 0,
(ii) dimV = n is even,
(iii) d i m l m / = ^n.
Suppose that t G £(C 5 ,C 5 ) is represented with respect to the basis
{(1,0,0,0,0), (1,1,0,0,0), (1,1,1,0,0), (1,1,1,1,0),(l, 1,1,1,1)}
by the matrix
1 8 6 4 1
0 1 0 0 0
0 1 2 1 0
0 -1 -1 0 1
0 -5 -4 -3 -2
Find a basis of C5 with respect to which the matrix of t is in Jordan
normal form.
Time allowed : 3 hours
(Allocate 20 marks for each question)
Let V be a finite-dimensional vector space. Prove that if / € £(V, V)
then
(a) dim V = dim Im / + dim Ker / ;
(b) the properties
(i) / is surjective,
(ii) / is injective,
are equivalent;
(c) V = Imfe Ker/ if and only if I m / = I m / 2 ;
(d) Im / = Ker / if and only if the following properties are satisfied
(i) f2 = 0,
(ii) dimV = n is even,
(iii) d i m l m / = ^n.
Suppose that t G £(C 5 ,C 5 ) is represented with respect to the basis
{(1,0,0,0,0), (1,1,0,0,0), (1,1,1,0,0), (1,1,1,1,0),(l, 1,1,1,1)}
by the matrix
1 8 6 4 1
0 1 0 0 0
0 1 2 1 0
0 -1 -1 0 1
0 -5 -4 -3 -2
Find a basis of C5 with respect to which the matrix of t is in Jordan
normal form.
