Let $T:\ V\rightarrow W$ be a linear transformation, if $\dim(V)=\dim(W)$, $\{v\_1,...,v\_n\}$ is a basis for $V$ and $\{w\_1,...,w\_n\}$ is a basis for $W$. Let $T:V\rightarrow W$ be a linear transformation defined by $T(v\_i)=w\_i,\ i=1,...n$. How to show $T$ is bijective?

Here is my try:

Since we know $\{w\_1,...w\_2\}$ is a basis for $W$ and $T(v\_i)=w\_i,\ i=1,...n$, $\{T(v\_1),...,T(v\_n)\}$ is a basis for $W$. $\{T(v\_1),...,T(v\_n)\}$ is linearly independent and(can't find an imply relationship) $\{v\_1,...,v\_n\}$ is independent. Then by $T(v\_i)=w\_i$, $T$ is bijective.

I found my process nonsense. Could someone help?