A solution of a linear system in some extension field implies a solution in the subfield

Question

Fix a field extension $k\subseteq K$ and consider a linear system $Ax=b$ where $A$ is a matrix (not necessarily square) with coefficients in $k$. I don't understand why if the above linear system has a solution in $K$, then there exists also a solution in $k$.

Thanks in advance.

Answer 1

There exists a $k$-linear projection $\pi:K\to k$. Applying $(\pi,\ldots,\pi)$ to both sides of $Ax=b$, we see that if $(x_1,\ldots,x_n)$ is a solution, so is $(\pi(x_1),\ldots,\pi(x_n))$.