Note Set 15e: The elation of projecting a vector onto a subspace

We deliver a more general kind of projection than the one we used for the Normal Equation. Given a basis for a subspace $\mathbf{S}$ in $R^m$, we show how to project a vector $\vec{b}$ onto that subspace. We produce a Projector Operator/Matrix showing how $\mathbf{A}^{\mbox{T}}\mathbf{A}$ has an inverse if and only if the Nullspace of $\mathbf{A}$ is zilch. The proof suggested by the monks is very enjoyable.
Best consumed by 2016/10/17