Coordinate Vector

Definition: Coordinate Vector

Let $B=({\mathbf{b}}_1,\cdots ,{\mathbf{b}}_n)$ be an ordered basis of a finitely generated vector space $(V,F,+,\cdot )$.

If $\mathbf{v}\in V$ has the basis representation $\mathbf{v}=v_1{\mathbf{b}}_1+\cdots +v_n{\mathbf{b}}_n$, then the coordinate vector of $\mathbf{v}$ with respect to the basis $B$ is the column vector

$${}_B\mathbf{v}\stackrel{\text{def}}{=}\left[\begin{array}{c}{v}_1\\ ⋮\\ v_n\end{array}\right]\in F^n$$