Product of matrix and vectors in orthogonal subspaces Let $p < n$, and let $A \in \mathbb{R}^{n \times n}$ be rank $p$. Let $V \in \mathbb{R}^{n \times n}$ be an orthogonal matrix with columns $V = \begin{bmatrix} v_1 & \cdots & v_n \end{bmatrix}$.
Suppose that for any linear combination of the first $p$ columns of $V$, that the product $A(c_1 v_1 + \cdots + c_p v_p) \ne 0$. Is it true that $A v_i = 0$ for all $i > p$?
I believe the answer to this is yes. By rank-nullity theorem, $A$ must have nullity $n-p$. Since $Aw \ne 0$ for any $w$ spanned by $v_1, \cdots, v_p$, and since the subspace ${\rm span}(v_1, \cdots ,v_p)$ is orthogonal to the subspace ${\rm span}(v_{p+1}, \cdots ,v_n)$, it must follow that $A v_i = 0$ for all $i > p$.
Any corrections or a more detailed proof would be much appreciated!

$A=\begin{pmatrix}1&0&1\\0&1&0\\0&0&0\end{pmatrix}$
$\operatorname{Rank} A=p=2$
$V=\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}$
Then $\begin{align}A(c_1v_1+c_2v_2) &=\begin{pmatrix}1&0&1\\0&1&0\\0&0&0\end{pmatrix}\begin{pmatrix}c_1\\c_2\\0\end{pmatrix}\\  &=\begin{pmatrix}c_1\\c_2\\0\end{pmatrix}\neq 0 [\text{ as $c\neq 0$ }]\end{align}$
$\begin{align}Av_3&=\begin{pmatrix}1&0&1\\0&1&0\\0&0&0\end{pmatrix}\begin{pmatrix}0\\0\\1\end{pmatrix}\\  &=\begin{pmatrix}1\\0\\0\end{pmatrix}\neq 0\end{align}$
