Trying to do this one:
Suppose $A$ is an invertible $n$ x $n$ matrix and the vectors $v_1$, $v_2$, ..., $v_n$ are linearly independent. Show that the vectors $Av_1$, $Av_2$, ..., $Av_n$ are linearly independent.
I know that A's column vectors are linearly independent since A is invertible. I also know there is no relation amongst the $v_i$ because they're linearly independent.
My idea is to write the product of A and a given $v_i$ in terms of the columns of A. Not sure if this is right, any guidance much appreciated!