Projection
A projection is an approximation within a column space.
Description
Given vectors a and b, a can be projected into the column space of b (i.e., C(b)). This projection p has an error term e.
If a is orthogonal to C(b), there is no projection.
Scalar Projection
A vector in the direction of b with the magnitude of a is given by ||b|| cos(θ) where θ is the angle formed by a and b. This can be called the scalar projection.
The dot product can be substituted into this definition to give a⋅b/||a|| or aTb/||a||.
Vector Projection
A vector projection is very similar to the scalar projection, but should have a magnitude based on how much a moved through C(b). This is captured by â, the unit vector in the direction of a, and is calculated as a/||a||.
Altogether, the vector projection could be given by any of:
||b|| cos(θ) â
(a⋅b/||a||) â
(aTb/||a||) â
Generally these reformulations are more useful:
p = (a⋅b/||a||) a/||a||
p = (a⋅b/||a||2) a
p = (a⋅b/a⋅a) a
...or:
p = (a⋅b/||a||) â
p = (â⋅b) â
The linear algebra view of this is that linear transformation from vector a to projection vector p is expressed as p = ax̂. The projection carries an error term that can be characterized by e = b - p or e = b - ax̂. a is orthogonal to e, so a⋅(b - ax̂) = 0. This simplifies to x̂ = (a⋅b)/(a⋅a). Again, the vector projection is given by p = a (a⋅b)/(a⋅a).