Vector Sub-Space

Def) Vector Sub-Space

Given a F.V.S. $V$, we consider a non-empty subset, we will denote as $U$, as a “Sub Space” if it is

1. Closed under addition
2. Closed under Scalar Multiplication Note : The closure of the binary operations in the Sub Space must be the same binary operations as the larger Vector Space

Consequences of Vector Sub-Spaces

If $U$ is a subspace of an F.V.S $V$, then:

1. If $0\in V,0\in U$
2. $\forall u\in U,\exists -u\in U$ such that $u+(-u)=0$

Proof

1. Pick any $u\in U$, since U is closed under scalar multiplication, then $0\cdot u=0,\forall u\in U$, therefore, $0\in U$
2. Pick any $u\in U$, then

The four fundamental sub-spaces of any vector space