# Linear Subspace

DEFINITION1: Let $K$ be a field, $X$ be a $K$-vector space and $M\subset X$. If $M$ is also a $K$-vector space, then $M$ is called a linear subspace or vector subspace (or shortly subspace) of $X$.

PROPOSITION1: Let $K$ be a field, $X$ be a $K$-vector space and $M\subset X$. $M$ is a subspace of $X$ if and only if

a) $\theta\in M$,

b) $x+y\in{M}$ for all $x,y\in{M}$,

c) $\lambda{x}\in{M}$ for all ${\lambda}\in{K}$ and ${x}\in{M}$.