Test paper 3
Time allowed : 3 hours
(Allocate 20 marks for each question)
If U, W are subspaces of a finite-dimensional vector space V prove that
dim U + dimW = d\m{U + W) + dim(tf n W).
Suppose now that V — U © W. If S is any subspace of V prove that
2 dim S - dim V < d\m[{U nS)®{Wn S)} < dim 5.
In the case where V = ©^ = 1 Ui find similar upper and lower bounds
for
dim0(C/tn5).
For the matrix
0 1 0
A= -1 1 1
-10 2
find a non-singular matrix P such that P~lAP is in Jordan normal
form.
If A is positive, obtain real values of z,-y such that C2 = D where
x13" "A 1 0*
C = 0 X22 ^23 0 A 1
0 0 ^33 0 0 A
Hence, or otherwise, find a real matrix X such that X2 = A.
Time allowed : 3 hours
(Allocate 20 marks for each question)
If U, W are subspaces of a finite-dimensional vector space V prove that
dim U + dimW = d\m{U + W) + dim(tf n W).
Suppose now that V — U © W. If S is any subspace of V prove that
2 dim S - dim V < d\m[{U nS)®{Wn S)} < dim 5.
In the case where V = ©^ = 1 Ui find similar upper and lower bounds
for
dim0(C/tn5).
For the matrix
0 1 0
A= -1 1 1
-10 2
find a non-singular matrix P such that P~lAP is in Jordan normal
form.
If A is positive, obtain real values of z,-y such that C2 = D where
x13" "A 1 0*
C = 0 X22 ^23 0 A 1
0 0 ^33 0 0 A
Hence, or otherwise, find a real matrix X such that X2 = A.
