Coordinate Vector (Linear Algebra)

Let $B = \{\vec{b_1}, \vec{b_2}, \ldots, \vec{b_n}\}$ be a basis for the vector space $V = \mathbb{R}^n$.

Then, every vector $\vec{x} \in V$ can be written uniquely as a linear combination:

$$x = c_1 \vec{b_1} + c_2 \vec{b_2} + \ldots + c_n \vec{b_n}$$

The vector

$$[\vec{x}]_B = \begin{pmatrix}c_1 \\ c_2 \\ \vdots \\ c_n \end{pmatrix}$$

is then called the coordinate vector[📖LLM21, p. 256] of $\vec{x}$ relative to B.

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References

1. [LLM21]: Lay, David and Lay, Steven and McDonald, Judi: Linear Algebra and Its Applications Global Edition (2021), Pearson Deutschland [BibTeX]