Definition:Coordinate Vector

Definition
Let $R$ be a ring with unity.

Let $M$ be a free $R$-module of dimension $n$.

Let $B = \left \langle {b_k} \right \rangle_{1 \mathop \le k \mathop \le n}$ be an ordered basis of $M$.

Let $x\in M$.

If $\lambda_1,\ldots,\lambda_n\in R$ are such that $x = \displaystyle\sum_{i=1}^n \lambda_i b_i$, then $(\lambda_1,\ldots,\lambda_n)^\intercal \in R^n$ is the coordinate vector of $x$ with respect to $B$.

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