Linear Algebra, 2020a

Section VI. Projection 275
VI
Projection
This section is optional. It is a prerequisite only for the final two sections
of Chapter Five, and some Topics.
We have described projection 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 ⎠
−1
So perhaps a better description is: the projection of ⃗v is the vector ⃗p in the plane
with the property that someone standing on ⃗p and looking straight up or down —
that is, looking orthogonally to the plane — sees the tip of ⃗v. In this section we
will generalize this to other projections, orthogonal and non-orthogonal.
VI.1
Orthogonal Projection Into a Line
We first consider orthogonal projection of a vector ⃗v into a line l. This shows
a figure walking out on the line to a point ⃗p such that the tip of ⃗v is directly
above them, where “above” does not mean parallel to the y-axis but instead
means orthogonal to the line.
Since the line is the span of some vector l = {c · ⃗s | c ∈ R}, wehaveacoefficient
c ⃗p with the property that ⃗v − c ⃗p ⃗s is orthogonal to c ⃗p ⃗s.
⃗v
⃗v − c ⃗p ⃗s
c ⃗p ⃗s