Linear Algebra

Section VI. Projection 263
N
M
Notice that the projection along N is not orthogonal—there are members of the
plane M that are not orthogonal to the dotted line. But the projection along
ˆN is orthogonal.
A natural question is: what is the relationship between the projection operation
defined above, and the operation of orthogonal projection into a line?
The second picture above suggests the answer—orthogonal projection into a
line is a special case of the projection defined above; it is just projection along
a subspace perpendicular to the line.
N
In addition to pointing out that projection along a subspace is a generalization,
this scheme shows how to define orthogonal projection into any subspace of R n ,
of any dimension.
3.4 Definition The orthogonal complement of a subspace M of R n is
M ⊥ = {�v ∈ R n � � �v is perpendicular to all vectors in M}
(read “M perp”). The orthogonal projection proj M (�v ) of a vector is its projection
into M along M ⊥ .
3.5 Example In R3 , to find the orthogonal complement of the plane
⎛
P = { ⎝ x
⎞
y⎠
z
� � 3x +2y − z =0}
we start with a basis for P .
ˆN
M
⎛
B = 〈 ⎝ 1
⎞ ⎛
0⎠
, ⎝
3
0
⎞
1⎠〉
2
Any �v perpendicular to every vector in B is perpendicular to every vector in the
span of B (the proof of this assertion is Exercise 19). Therefore, the subspace
M