The title is rather vague, I know - I'm struggling to understand this concept (let alone explain it succinctly) and would really appreciate some help.
I'm working through a pre-test for an exam tomorrow, and I'm stuck on how to prove this one - in class we've reached the beginning of the Invertible Matrix Theorem.
Let $A$ be an $m \times n$ matrix and let $T : \mathbb R^m \to\mathbb R^k$ be a linear transformation. Show that if $v$ is a solution of $Ax = b$ then it is also a solution of $Bx = T(b)$ where $B = [T(a_1)\ T(a_2) \cdots T(a_n)]$ and $\{a_i\}$ are the columns of $A$.
So far, I've gathered that it will have something to do with the fact that the transform can be applied communally - I think - but I really can't even figure out a first step! What would you recommend - how do I prove this?