Vector Space
Subspace of a Vector Space
- It contains Zero vector i.e $Zero_v \in W$
- closed under vector addition i.e $\forall u,v \in W$, $u+v \in W$
- closed under scalar multiplication i.e $\forall v \in W$ and $\forall a \in F$ , $a.v \in W$
Every subspace of a vector space is a vector space
Example:- The set of all Diagonal Matrices, The set of all square matrices $M_{nxn}$ having trace = 0 ( You can prove that using the fact that $\forall M,N \in M_{nxn}$ $Tr(a.M) = a.Trace(M)$ and $Tr(M+N) = Tr(M) + Tr(N)$ )
Non-Example:- Set of all Matrices having non-negative entries (Violates Rule 3)
- Any intersection of Subspaces of a Vector Space is a Vector Space i.e if $C = $ {all possible subspace, $W$ of $V$ s.t $W \subset V$ }, then $\cap W$ is also a Vector Space.