B-coordinate

If we have some ordered basis of a subspace $V$ $\beta =\{{\vec{v}}_1,\dots ,{\vec{v}}_n\}$, the $\beta$-coordinates of $\vec{v}\in V$ are the unique scalars such that:

$$\vec{v}=a_1{\vec{v}}_1+\cdots +a_n{\vec{v}}_n$$

We usually represent the scalars as a column vector $[\vec{v}{]}_{\beta }$.

$$[\vec{v}{]}_{\beta }=\left[\begin{array}{c}{a}_1\\ ⋮\\ a_n\end{array}\right]$$