Ordered Bases and Coordinate Matrices
- Linear Algebra 1.5
Ordered Basis
Let $V$ be a vector space of dimension $n$ on a field $F$. An ordered basis for $V$ is an ordered $n$-tuple of vectors $(\b{v}_1, \dots, \b{v}_n)$ such that $\Set{\b{v}_1, \dots, \b{v}_n}$ is a basis of $V$. If $\mathcal{B} = (\b{v}_1, \dots, \b{v}_n)$ is an ordered basis of $V$, then for each vector $\b{v} \in V$, there exist unique ordered $n$-tuples of scalars $(a_1, \dots, a_n)$ such that
\[ \b{v} = \sum_{i=1}^n a_i \b{v}_i \]
Coordinate Map and Coordinate Matrix
Accordingly, we can define the coordinate map $\phi_\mathcal{B} : V \to F^n$ by
\[ \phi_\mathcal{B}(\b{v}) = [\b{v}]_\mathcal{B} = \begin{bmatrix} a_1 \nl \vdots \nl a_n \end{bmatrix} \]
where the column matrix $[\b{v}]_\mathcal{B}$ is called the coordinate matrix of $\b{v}$ relative to $\mathcal{B}$. Clearly, knowing $\mathcal{B}$ and $[\b{v}]_\mathcal{B}$ allows us to reconstruct $\b{v}$.
Furthermore, it is easy to see that the coordinate map $\phi_\mathcal{B}$ is bijective and preserves the vector space operations, that is:
\[ \phi_\mathcal{B}(\b{v} + \b{w}) = \phi_\mathcal{B}(\b{v}) + \phi_\mathcal{B}(\b{w}) \nl \phi_\mathcal{B}(c \b{v}) = c \phi_\mathcal{B}(\b{v}) \]
