Definition:Coordinate Vector

Definition
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: $\sqbrk x_B$.

Also see

 * Expression of Vector as Linear Combination from Basis is Unique, which justifies this definition
 * Change of Coordinate Vector Under Change of Basis
 * Definition:Change of Basis Matrix