Let $A$ be an $n\times n$-matrix. Three column vectors $X_1,X_2,X_3$ are given, and we have that $$A^2X_1=A^2X_2=A^2X_3=0$$ We know that $AX_1,AX_2, AX_3$ are linearly independent. Prove that the column vectors $X_1,X_2,X_3,AX_1,AX_2,AX_3$ are also linearly independent.
It is a hard problem for me, and my first thought was that $A^2X_1=0 \Rightarrow AX_1=0$. That is probably not correct and not leading to anything.
I then concluded that $\det(AX_1,AX_2, AX_3)\not=0$ which can lead me somewhere, since it's a proof with linear independence and $\det\not=0 \Leftrightarrow$ lin. independent.