A basis for a vector space $V$ is a collection of vectors that spans $V$ and is linearly independent. The wonderful thing about bases is that they provide coordinate systems for our vector spaces.

If $\mathcal{B} = \{{\bf b}_1, {\bf b}_2, \ldots, {\bf b}_n\}$ is a basis for $V$, then every vector ${\bf v} \in V$ can be expressed as a linear combination $${\bf v} = a_1 {\bf b}_1 +a_2 {\bf b}_2 + \cdots + a_n {\bf b}_n$$ in exactly one way. The numbers $a_1, \ldots, a_n$ are called the coordinates of ${\bf v}$ in the $\mathcal{B}$ basis, and we write $$ [{\bf v}]_{\mathcal{B}} = \begin{pmatrix} a_1 \cr a_2 \cr \vdots \cr a_n \end{pmatrix}.$$

Using coordinates makes $V$ behave just like ${\bf R}^n$. If we have a bunch of vectors in $V$, we can tell whether they are linearly independent, whether they span, or whether they have other properties, by taking their coordinates and seeing whether the coordinate vectors have those properties in ${\bf R}^n$. That's something we learned to do in the previous section.