Now it's all happening. We work through an example showing how square matrices have certain preferred directions: eigenvectors. We set up one of the most beautiful equations in all the land: $(\mathbf{A} - \lambda\mathbf{I})\vec{v} = \vec{0}$. Eigenvectors, eigenvalues, and eigenspaces will be the big pieces and our old friend nullspace reappears. It's a long episode so dig in and absorb it all. We hear that determinants, a numinous monk legend, will helps us solve the eigenvalue equation, and we head off in that direction next.