Large Matrix Dimension Generalization

Question

Let $X$ and $Y$ be two $7\times 6$-matrices, such that $Xv = Yv$ for all vectors $v$ in $\mathbb{R}^6$.

How can I show, if possible, that

1. $\mathrm{rank}(A) = \mathrm{rank}(B)$
2. $\mathrm{rref}(A) = \mathrm{rref}(B)$
3. $A = B$

Without having to construct a tediously large $7\times6$-matrix, how can I generalize to a more convenient matrix to confirm the validity of these claims?

Thank You

Answer 1

We can show that $A = B$. Let $e_1,\dots,e_6$ denote the standard basis (so for example, $e_2 = (0,1,0,\dots,0)^T$). Note that $Ae_k$ is the $k$th column of $A$. Since $Ae_k = Be_k$ for all $k$, $A$ and $B$ have the same columns. Since they have the same columns, they must be the same matrix.