If we have two linear transformations denoted by matrices $A, B$ operating on an arbitrary vector $v \in \mathbb R^n$, then how does $Av$ and $Bv$ differ geometrically from $(A-B)v$ ? Does the difference inherit properties from the two linear transformations, or is there no pattern at all?
The reason why I ask pertains to eigen vectors, for they are sent to the null vector when operated upon by $\lambda I - A$ and I am trying to geometrically understand why. If you can explain this second question and not the first one, that would also be fine.
One motivation for the concept of eigenvectors and eigenvalues is the following question: “Does a given linear transformation map some line through the origin onto itself?” If there is such a line for the linear transformation $A$, then for every vector $\mathbf v$ on that line we must have $A\mathbf v=\lambda \mathbf v$, where $\lambda$ is a fixed scalar. So, if $\mathbf v$ is an eigenvector corresponding to the eigenvalue $\lambda$, then $(A-\lambda I)\mathbf v=0$ simply says that $A$ maps $\mathbf v$ to another vector on the line through the origin that contains $\mathbf v$ itself.