Automorphism Group of Vector Spaces

Theorem

Let $V$ be a vector space, then set $Aut(V)$ is also a Group

Proof

1. If $S,T\in Aut(V)$ then, $ST\in Aut(g)$
2. The associative multiplication operation is composition
3. The identity transformation $I\in Aut(g)$ is the identity transformation. $T(I(v))=T(v)=I(T(v))$ for all $v\in V$
4. If $T\in Aut(V)$ then, $T$ is invertable by definition of Automorphism with inverse $T^{-1}\in Aut(V)$