Linear Algebra

250 Chapter 3. Maps Between Spaces
3.VI Projection
The prior section describes the matrix equivalence canonical form as expressing a
projection and so this section takes the natural next step of studying projections.
However, this section is optional; only the last two sections of Chapter Five
require this material. In addition, this section requires some optional material
from the subsection on length and angle measure in n-space.
We have described the projection π from R 3 into its xy plane subspace as
a ‘shadow map’. This shows why, but it also shows that some shadows fall
upward.
� �
1
2
2
� �
1
2
0
� �
1
2
0
� �
1
2
−1
So perhaps a better description is: the projection of �v is the �p in the plane with
the property that someone standing on �p and looking straight up or down sees
�v. In this section we will generalize this to other projections, both orthogonal
(i.e., ‘straight up and down’) and nonorthogonal.
3.VI.1 Orthogonal Projection Into a Line
We first consider orthogonal projection into a line. To orthogonally project
a vector �v into a line ℓ, darken a point on the line if someone on that line and
looking straight up or down (from that person’s point of view) sees �v.
�v
The picture shows someone who has walked out on the line until the tip of
�v is straight overhead. That is, where the line is described as the span of
some nonzero vector ℓ = {c · �s � � c ∈ R}, the person has walked out to find the
coefficient c�p with the property that �v − c�p · �s is orthogonal to c�p · �s.
�p
ℓ