Answer Key to Second Midterm Examination, Version 1
1.
Let
V
be a two dimensional subspace of
5
. Answer the following questions (a correct
answer without step by step description of your reasoning: maximum
10
points
deduction
each):
(a) Find the dimension of
V
.
The space
V
is the kernel of
proj
V
. Since Im
proj
V
V
, rank
proj
V
dim
V
2.
Thus the nullity is 5
2
3; so, the dimension of
V
is 3.
(b) Prove that
V
V
0
.
if
x
x
1
x
2
.
.
.
x
5
is in
V
V
, then
x
2
1
x
2
2
x
2
3
x
2
4
x
2
5
x
x
0 because the left
x
can be considered to be in
V
and the right
x
can be considered to be in
V
. Since
x
2
j
0 for all
j
1 2
5, the only possibility for
x
j
to have
x
2
1
x
2
2
x
2
5
0 is
x
1
x
2
x
5
0. Thus
x
0 and
V
V
0
.
(c) For two distinct orthonormal basis
v
1
v
2
and
u
1
u
2
of
V
and any given
vector
x
in
5
, let
v
v
1
x
v
1
v
2
x
v
2
and
w
u
1
x
u
1
u
2
x
u
2
. Prove that
v
w
. Hint: Show first that
x
v
is in
V
, and use this fact and
(b) to show
v
w
.
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 Fall '08
 lee
 Linear Algebra, Algebra, Vector Space, dimensional subspace

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