Suppose $\{v_1,v_2,\ldots,v_n\}$ is a linearly independent set of vectors of the vector space $V$ and $A$ is an $n×n$ matrix. Let
$[u_1 u_2 \ldots u_n]^T=A [v_1 v_2 \ldots v_n]^T$
Show that $\{u_1,u_2,\ldots,u_n\}$ is linearly independent if and only if $A$ is invertible.
I have proved that if $A$ is invertible, then {${u_1,u_2,\ldots,u_n}$} is linearly independent, but I can not show the converse.
And I want to mention this problem is an exercise from a book that posed this in the vector space section before introducing basis and dimension of spaces. So please help maintaining this.