prove that $\{A_1,...,A_m\}$ cannot span $M_{nn}$ if $\vec{0} \neq y \in \mathbb{R}$ and $A_1y=A_2y=...=A_my=0$

Question

Let $A_1, A_2,...,A_m$ denote n by n matrices if $\vec{0} \neq y \in \mathbb{R}$ and $A_1y=A_2y=...=A_my=0$, prove that $\{A_1,...,A_m\}$ cannot span $M_{nn}$

My Proof (Edited)

Let $B \in M_{nn}$ and invertible. Now assume that $M_{nn} = span\{A_1, A_2,...,A_m\}$

$$B = c_1A_1 + ...+c_mA_m$$ $$By = [c_1A_1 + ...+c_mA_m]y$$ $$By = c_1A_1y + ...+c_mA_my = c_10+...+c_m0=0$$

Hence $By=0$. Since it is invertible $y = B^{-1}0 = 0$ which is a contradiction.

Is this approach (now) correct?

Answer 1

"Hence...either..." is false.

In general, $Ax=0$ does not imply that either $A=0$ or $x=0$.

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Hint for a proof.

Let $B$ be an invertiable matrix of $M_{nn}$ and assume that $B$ is a linear combination of the matrices $A_k$. Calculate $By$.