The linear transformation $T : \Bbb R^4 \rightarrow \Bbb R^4$ has $2$ dimensional range, and is represented by matrix $A$.
Let $M$ be a given $4 × 4$ matrix and let $S$ be the vector space consisting of vectors of the form $MAx$, where $x \in R^4$.
Show that if $M$ is non-singular then the dimension of $S$ is $2$.
My attempt at a solution :
$Ax = b = a_1v_1 + a_2v_2$, where $(v_1,v_2)$ are a basis of $T$. Applying $M$, we get
$MAx = Mb = M(a_1v_1) +M(a_2v_2)$
$MAx = a_1(Mv_1) + a_2 (Mv_2)$
How do I use the fact that $M$ is non-singular to prove that $Mv_1$ and $Mv_2$ are linearly independent?