# Let Mn,p(K) be the set of matrices n×p with coefficients in K. Let A in Mn,p(Q). We suppose there exists a non zero solution X in Mp,1(R) to AX=0. (0 denotes [0]p,1) Show that there exists a non zero solution X′ in p,1(Q) to AX′=0

Cristofer Watson 2022-10-16 Answered
Let ${\mathcal{M}}_{n,p}\left(\mathbb{K}\right)$ be the set of matrices $n×p$ with coefficients in $\mathbb{K}$.
Let $A\in {\mathcal{M}}_{n,p}\left(\mathbb{Q}\right)$.
We suppose there exists a non zero solution $X\in {\mathcal{M}}_{p,1}\left(\mathbb{R}\right)$ to $AX=0$. ($0$ denotes $\left[0{\right]}_{p,1}$)
Show that there exists a non zero solution ${X}^{\prime }\in {}_{p,1}\left(\mathbb{Q}\right)$ to $A{X}^{\prime }=0$
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