Test paper 2
Time allowed : 3 hours
(Allocate 20 marks for each question)
1 Let 7 be a finite-dimensional vector space and let e G JC(V,V) be a
projection. Prove that
Kere = Im(idv —e).
If t G £(V, V) show that Im e is tf-invariant if and only if eotoe = to e;
and that Kere is ^-invariant if and only if eoto e = eot. Deduce that
Im e and Ker e are ^-invariant if and only if e and t commute.
2 If U = [ur8] G Mat n X f l (C) is given by
fl ifs = r + l ;
r
* [0 otherwise,
and J - [jr9] G M a t n x n ( C ) is given by
hs =
\ 0 otherwise,
show that U1 — JUJ. Deduce that if A G M a t n x n ( C ) then there is an
invertible matrix P such that P"lAP = A1.
Find such a matrix P when A is the matrix
0 4 4
2 2 1
-3 -6 -5
Time allowed : 3 hours
(Allocate 20 marks for each question)
1 Let 7 be a finite-dimensional vector space and let e G JC(V,V) be a
projection. Prove that
Kere = Im(idv —e).
If t G £(V, V) show that Im e is tf-invariant if and only if eotoe = to e;
and that Kere is ^-invariant if and only if eoto e = eot. Deduce that
Im e and Ker e are ^-invariant if and only if e and t commute.
2 If U = [ur8] G Mat n X f l (C) is given by
fl ifs = r + l ;
r
* [0 otherwise,
and J - [jr9] G M a t n x n ( C ) is given by
hs =
\ 0 otherwise,
show that U1 — JUJ. Deduce that if A G M a t n x n ( C ) then there is an
invertible matrix P such that P"lAP = A1.
Find such a matrix P when A is the matrix
0 4 4
2 2 1
-3 -6 -5
