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}.

» Read more