Note Set 25a: The Fundamental Theorem of Matrixology redux

Our epic quest to understand $\mathbf{A}\vec{x} = \vec{b}$ ends with Singular Value Decomposition allowing us to finish writing the Big Picture story. Row space and Column space and Nullspace and Left Nullspace all have a natural, best basis derived from the eigenvectors of $\mathbf{A}^{\rm T}\mathbf{A}$ and $\mathbf{A}\mathbf{A}^{\rm T}$. While this will undoubtedly feel like an exhilarating conclusion, it's really just a beginning. The monks wave you on.
Best consumed by 2016/12/05