Prove linear independence Prove from the definition of "linearly independent" that if ${\{v_1,v_2,...,v_n\}}$ is linearly independent and if A is an invertible $n x n$ matrix, then the set ${\{Av_1,Av_2,...,Av_n\}}$ is linearly independent.
I ran in to this problem in my textbook and I am having trouble understanding how to reach the answer. My initial thought is that is has something to do with the fact that for a matrix to be invertible it's determinant must be non-zero. But that seems to contradict the definition of linearly independent which states that the only scalars that make $c_1v_1+c_2v_2+...+c_nv_n=0$ true must be $c_1=c_2=...=c_n = 0$

Suppose that 
$$
\lambda_1\, Av_1+\lambda_2\, Av_2+\dotsb+\lambda_n\, Av_n=\vec 0
$$
Then
$$
A\bigl(
\lambda_1v_1+\lambda_2v_2+\dotsb+\lambda_nv_n
\bigr)
=
\vec 0\tag{1}
$$
Since $A$ is invertible we know that $A^{-1}$ exists. What happens when we multiply both sides of (1) on the left by $A^{-1}$?
