# Definition:Coordinate Vector

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## Definition

Let $R$ be a ring with unity.

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

Let $B = \sequence {b_k}_{1 \mathop \le k \mathop \le n}$ be an ordered basis of $M$.

Let $x\in M$.

Let $\lambda_1, \ldots, \lambda_n\in R$ be such that $\ds x = \sum_{i \mathop = 1}^n \lambda_i b_i$.

Then $\tuple {\lambda_1, \ldots, \lambda_n} \in R^n$ 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

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