Coordinate Vector

Topics: Algebra - Vector Space

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Let $V$ be a vector space with $\dim (V)=n$. Let $\beta =\{x_1,\dots ,x_n\}$ be a basis of $V$.

Given $x\in V$, we define the coordinate vector of $x$ relative to the basis $\beta$ as:

$$[x{]}_{\beta }=\left(\begin{array}{c}{a}_1\\ a_2\\ \dots \\ a_n\end{array}\right)$$

…such that $x={\sum }_{i=1}^n{a}_i{x}_i$.

That is, the elements in the coordinate vector are the coefficients of $x$ represented as a linear combination of the elements in $\beta$).