Let $\mathbb K$ a commutative field and $\mathbb L$ an extension field of $\mathbb K$ (that is to say a field that contains $\mathbb K$).
Let $A$ be a $n \times p$ matrix over $\mathbb K$.
Let $B$ be a $n \times 1$ matrix over $\mathbb K$.
Let $S_k=\{X,X \; \text{is a} \;p \times 1 \; \text{matrix over } \; \mathbb K \; \text{such that} \; AX=B \}$
Let $S_l=\{X,X \; \text{is a} \;p \times 1 \; \text{matrix over } \; \mathbb L \; \text{such that} \; AX=B \}$
Prove that $$S_k \neq \emptyset \Leftrightarrow S_l \neq \emptyset$$
Clearly the left-to-right implication is true. I have no clue how to do the converse though. Any help appreciated.
Let $S_l\ne\varnothing$. It is sufficiently to consider the case when $n<p$ and ${\rm rank} A=n$. One can to choose values of $n$ unknowns arbitrary from $\mathbb{K}$ (see wikipedia). Then others are expressed through them and coefficients of $A$, so they also belong to $\mathbb{K}$.