System of linear equations/Set of variables/Solution space is vector space/Fact/Proof/Exercise

Let ${}K$ be a field, and let

${\begin{matrix}a_{11}x_{1}+a_{12}x_{2}+\cdots +a_{1n}x_{n}&=&0\\a_{21}x_{1}+a_{22}x_{2}+\cdots +a_{2n}x_{n}&=&0\\\vdots &\vdots &\vdots \\a_{m1}x_{1}+a_{m2}x_{2}+\cdots +a_{mn}x_{n}&=&0\end{matrix}}$

be a system of linear equations over ${}K$. Show that the set of all solutions of this system is a linear subspace of ${}K^{n}$. How is this solution space related to the solution spaces of the individual equations?

Create a solution