Suppose $v_1,\dots,v_n$ is a basis of $V$. Prove that the map $T : V \to \mathbf F^{n,1}$ defined by

is an isomorphism of $V$ onto $\mathbf F^{n,1}$; here $\mathcal M(v)$ is the matrix of $v \in V$ with respect to the basis $v_1,\dots,v_n$.

TODO