Let $V$ and $W$ be vector spaces over a field $K$ and $T : V \to W$ be a linear map
Suppose that $V = W$. Show that if $T$ is injective or surjective, then $T$ is an
isomorphism
Injective: $\ker(T)=\{0\}$, therefore $\{v \in V:T(v)=0\}=\{0\}$. Thus a one to one mapping from $0$ to $0$.
Since $V=W$, we have the same dimension, therefore the same mapping, therefore for all points there exists a one to one mapping.
Surjective: $\mathrm{Im}(T)=V$, since the image is the whole set then, $w\in V:T(v)=w$ is the whole of the codomain, since they are equivalent, we have a one to one mapping from the domain to the codomain, hence a bijection.
Does this logic make sense or is a good enough answer, have I missed anything.
Any help will be greatly appreciated, thanks.
Best Answer
We have $$T\;\text{is injective}\iff \ker T=\{0\}$$ and
$$T\;\text{is surjective}\iff \operatorname{im} T=V$$ and we see by the rank-nullity theorem (in finite dimensional space) that
$$T\;\text{is injective}\iff T\;\text{is surjective}\iff T\;\text{is bijective}$$