If $A$ is a square matrix that is linearly independent, is $AA$? I'm just not sure how to start this problem from Linear Algebra Done Wrong. The problem is to prove that if the columns of $A$, square matrix, are linearly independent, then the columns of $A^2$ = $AA$ are also linearly independent. I'm mostly just not sure how to start this proof.

Since $A$ is linearly independent, no (non-zero) linear combination of the rows of $A$ result in 0.  This means that for any non-zero vector $x$, $Ax\neq 0$.  The rows of $A^2$ are linearly independent if and only if there is no (non-zero) linear combination of them that results in zero, i.e. no nonzero $x$ such that $A^2 x=0$.  Then, view $(AA)x$ as $A(Ax)$.  Since $Ax\neq 0$, $A(Ax)\neq 0$.  This is true for any non-zero vector $x$, so the rows of $A^2$ are linearly independent.
