I have a homework problem that asks me to prove if $T(B)$ is a basis for $W$, then $T$ is an isomorphism. Here is the full question:

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> Let $V$ be a vector space of dimension $n$ over a field $F$ and let $B=\{v\_{1},v\_{2},\ldots,v\_{n}\}$ be a basis for $V$. Let $W$ be a vector space over $F$. Let $T \in\hom(V,W)$ such that $T(B)=\{T(v\_{1}),T(v\_{2}),...,T(v\_{n})\}$ is a set of $n$ distinct vectors in $W$. Prove that if $T(B)$ is a basis for $W$, then $T$ is an isomorphism.
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I know if I prove $\ker(v)=${0} (zero vector)then it is one to one. Hence it also proves onto because of dimension theorem. However, I cannot prove it is one to one. Please help me out.