Exercise 18

Does every correspondence between bases, when extended to the spaces, give an isomorphism? That is, suppose that $V$ is a vector space with basis $B=\langle\vec{\beta}_{1},\ldots,\vec{\beta}_{n}\rangle$ and that $f\colon B\to W$ is a correspondence such that $D=\langle f(\vec{\beta}_{1}),\ldots,f(\vec{\beta}_{n})\rangle$ is basis for $W$. Must $\hat{f}\colon V\to W$ sending $\vec{v}=c_{1}\vec{\beta}_{1}+\cdots+c_{n}\vec{\beta}_{n}$ to $\hat{f}(\vec{v})=c_{1}f(\vec{\beta}_{1})+\cdots+c_{n}f(\vec{\beta}_{n})$ be an isomorphism?