Linear Algebra, 2020a

288 Chapter Three. Maps Between Spaces
N
M
ˆN
M
Notice that the projection along N is not orthogonal since there are members
of the plane M that are not orthogonal to the dotted line. But the projection
along ˆN is orthogonal.
We have seen two projection operations, orthogonal projection into a line as
well as this subsections’s projection into an M and along an N, and we naturally
ask whether they are related. The right-hand picture above suggests the answer —
orthogonal projection into a line is a special case of this subsection’s projection;
it is projection along a subspace perpendicular to the line.
N
M
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 R 3 , to find the orthogonal complement of the plane
⎛ ⎞
x
⎜ ⎟
P = { ⎝y⎠ | 3x + 2y − z = 0}
z
we start with a basis for P.
⎛ ⎞ ⎛
1
⎜ ⎟ ⎜
0
⎞
⎟
B = 〈 ⎝0⎠ , ⎝1⎠〉
3 2
Any ⃗v perpendicular to every vector in B is perpendicular to every vector in the
span of B (the proof of this is Exercise 22). Therefore, the subspace P ⊥ consists