\begin{definition}[Definition:Coordinate Vector]
Let $R$ be a ring with unity.
Let $M$ be a free $R$-module of dimension $n$.
Let $B = \sequence {b_k}_{1 \mathop \le k \mathop \le n}$ be an ordered basis of $M$.
Let $x\in M$.
If $\lambda_1, \ldots, \lambda_n\in R$ are such that $\ds x = \sum_{i \mathop = 1}^n \lambda_i b_i$, then $\tuple {\lambda_1, \ldots, \lambda_n}^\intercal \in R^n$ is the '''coordinate vector''' of $x$ with respect to $B$.
This can be denoted: $\sqbrk x_B$.
\end{definition}