ASSIGNMENT 8
Solutions
Let
V
be the vector space of 3
×
3 matrices.
1.
[4 marks]
Give an example of a vector
v
∈
V
, and bases
B
and
C
for a subspace
U
⊆
V
, such that
[
v
]
B
=
0
1
0
and
[
v
]
C
=
1
0

2
.
The number of possible answers is uncountably infinite. Note that
B
and
C
will each have three elements.
Take, for example,
B
=
1
0
0
0
0
0
0
0
0
,
1
0

2
0
0
0
0
0
0
,
0
0
1
0
0
0
0
0
0
and
C
=
1
0
0
0
0
0
0
0
0
,
0
1
0
0
0
0
0
0
0
,
0
0
1
0
0
0
0
0
0
.
That
C
is linearly independent is obvious; that
B
is linearly independent is easily shown. Then let
v
=
1
0

2
0
0
0
0
0
0
.
2.
[3 marks]
Find
P
C
←
B
and
P
B
←
C
for your bases
B
and
C
from the previous question.
Clearly we have
P
C
←
B
=
1
1
0
0
0
1
0

2
0
.
Inverting this yields
P
B
←
C
=
1
0
1
/
2
0
0

1
/
2
0
1
0
.
Let
u
and
v
be vectors in
R
n
.
3.
[3 marks]
Suppose neither
u
, nor
v
is equal to
0
, and let
u
=
u
u
and
v
=
v
v
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 All
 Linear Algebra, Matrices, Vector Space, Equals sign, Hilbert space

Click to edit the document details