Math 215 HW #7 Solutions
1. Problem 3.3.8. If
P
is the projection matrix onto a
k
dimensional subspace
S
of the whole
space
R
n
, what is the column space of
P
and what is its rank?
Answer:
The column space of
P
is
S
. To see this, notice that, if
~x
∈
R
n
, then
P~x
∈
S
since
P
projects
~x
to
S
. Therefore, col(
P
)
⊂
S
. On the other hand, if
~
b
∈
S
, then
P
~
b
=
~
b
, so
S
⊂
col(
P
). Since containment goes both ways, we see that col(
P
) =
S
.
Therefore, since the rank of
P
is equal to the dimension of col(
P
) =
S
and since
S
is
k

dimensional, we see that the rank of
P
is
k
.
2. Problem 3.3.12. If
V
is the subspace spanned by (1
,
1
,
0
,
1) and (0
,
0
,
1
,
0), ﬁnd
(a)
a basis for the orthogonal complement
V
⊥
.
Answer:
Consider the matrix
A
=
±
1 1 0 1
0 0 1 0
²
.
By construction, the row space of
A
is equal to
V
. Therefore, since the nullspace of
any matrix is the orthogonal complement of the row space, it must be the case that
V
⊥
= nul(
A
). The matrix
A
is already in reduced echelon form, so we can see that the
homogeneous equation
A~x
=
~
0 is equivalent to
x
1
=

x
2

x
4
x
3
= 0
.
Therefore, the solutions of the homogeneous equation are of the form
x
2

1
1
0
0
+
x
4

1
0
0
1
,
so the following is a basis for nul(
A
) =
V
⊥
:

1
1
0
0
,

1
0
0
1
.
(b)
the projection matrix
P
onto
V
.
Answer:
From part (a), we have that
V
is the row space of
A
or, equivalently,
V
is the
column space of
B
=
A
T
=
1 0
1 0
0 1
1 0
.
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