Linear Algebra

Section VI. Projection 261
This definition doesn’t involve a sense of ‘orthogonal’ so we can apply it to
spaces other than subspaces of an R n . (Definitions of orthogonality for other
spaces are perfectly possible, but we haven’t seen any in this book.)
3.2 Example The space M2×2 of 2×2 matrices is the direct sum of these two.
�
a
M = {
0
�
b ��
a, b ∈ R}
0
�
0
N = {
c
�
0 ��
c, d ∈ R}
d
To project
A =
� �
3 1
0 4
into M along N, we first fix bases for the two subspaces.
�
1
BM = 〈
0
� �
0 0
,
0 0
�
1
〉
0
�
0
BN = 〈
1
� �
0 0
,
0 0
�
0
〉
1
The concatenation of these
�
⌢ 1
B = BM BN = 〈
0
� �
0 0
,
0 0
� �
1 0
,
0 1
� �
0 0
,
0 0
�
0
〉
1
is a basis for the entire space, because the space is the direct sum, so we can
use it to represent A.
� � � � � � � � � �
3 1 1 0 0 1 0 0 0 0
=3· +1· +0· +4·
0 4 0 0 0 0 1 0 0 1
Now the projection of A into M along N is found by keeping the M part of this
sum and dropping the N part.
� � � � � � � �
3 1 1 0 0 1 3 1
projM,N( )=3· +1· =
0 4 0 0 0 0 0 0
3.3 Example Both subscripts on projM,N(�v ) are significant. The first subscript
M matters because the result of the projection is an �m ∈ M, and changing
this subspace would change the possible results. For an example showing that
the second subscript matters, fix this plane subspace of R3 and its basis
⎛
M = { ⎝ x
⎞
y⎠
z
� ⎛
� y − 2z =0} BM = 〈 ⎝ 1
⎞ ⎛
0⎠
, ⎝
0
0
⎞
2⎠〉
1
and compare the projections along two different subspaces.
⎛
N = {k ⎝ 0
⎞
0⎠
1
� ⎛
� k ∈ R} N ˆ = {k ⎝ 0
⎞
1 ⎠
−2
� � k ∈ R}