# Linear Subspace

**DEFINITION1:** Let be a field, be a -vector space and . If is also a -vector space, then is called a linear subspace or vector subspace (or shortly subspace) of .

**PROPOSITION1:** Let be a field, be a -vector space and . is a subspace of if and only if

**a)** ,

**b)** for all ,

**c)** for all and .