Let ${\mathcal{M}}_{n,p}(\mathbb{K})$ be the set of matrices $n\times p$ with coefficients in $\mathbb{K}$.

Let $A\in {\mathcal{M}}_{n,p}(\mathbb{Q})$.

We suppose there exists a non zero solution $X\in {\mathcal{M}}_{p,1}(\mathbb{R})$ to $AX=0$. ($0$ denotes $[0{]}_{p,1}$)

Show that there exists a non zero solution ${X}^{\prime}\in {}_{p,1}(\mathbb{Q})$ to $A{X}^{\prime}=0$

Let $A\in {\mathcal{M}}_{n,p}(\mathbb{Q})$.

We suppose there exists a non zero solution $X\in {\mathcal{M}}_{p,1}(\mathbb{R})$ to $AX=0$. ($0$ denotes $[0{]}_{p,1}$)

Show that there exists a non zero solution ${X}^{\prime}\in {}_{p,1}(\mathbb{Q})$ to $A{X}^{\prime}=0$