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!
Thanks,
Mariogs
Best Answer
Let $w_i=Av_i$ for $i=1,\dots,n$. To show linear independence, we must show that $$ c_1w_1+\cdots+c_nw_n=0 \Longrightarrow c_i=0,\quad i=1,\dots n.$$ Since $A$ is invertible, we can left-multiply $c_1w_1+\dots=0$ by $A^{-1}$ to get...
Can you take it from there? Remember that the vectors $v_1,\dots v_n$ are linearly independent.