Linear Algebra/Projection

This section is optional; only the last two sections of Chapter Five require this material.

We have described the projection $$ \pi $$ from $$\mathbb{R}^3$$ into its $$ xy $$ plane subspace as a "shadow map". This shows why, but it also shows that some shadows fall upward. So perhaps a better description is: the projection of $$\vec{v}$$ is the $$\vec{p}$$ in the plane with the property that someone standing on $$\vec{p}$$ and looking straight up or down sees $$\vec{v}$$. In this section we will generalize this to other projections, both orthogonal (i.e., "straight up and down") and nonorthogonal.