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Let $V$ be a vector space and $B$ a basis for $V$. Say $B=\{x_1, \dots, x_n\}$ and consider $B$ as an ordered set. Since $B$ is a spanning set, then there exists a linearly independent combination of $x_1, \dots, x_n$ such that for any $x\in V$, we have $x=\alpha_1x_1 +\dots + \alpha_nx_n $ for some unique $\alpha_1, \dots, \alpha_n$. Then the vector
$$\begin{align*}
\begin{bmatrix}\alpha_1 \\ \dots \\ alpha_n\end{bmatrix} = [x]_{\beta}
\end{align*}$$ is the coordinate vector of $x$ with respect to $B$.
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Let $V$ be a vector space and $B$ a basis for $V$. Say $B=\{x_1, \dots, x_n\}$ and consider $B$ as an ordered set. Since $B$ is a spanning set, then there exists a linearly independent combination of $x_1, \dots, x_n$ such that for any $x\in V$, we have $x=\alpha_1x_1 +\dots + \alpha_nx_n $ for some unique $\alpha_1, \dots, \alpha_n$. Then the vector
$$\begin{align*}
\begin{bmatrix}\alpha_1 \\ \dots \\ alpha_n\end{bmatrix} = [x]_{\beta}
\end{align*}$$ is the coordinate vector of $x$ with respect to $B$.
Concepts
Coming soonUsed In
Coming soonHypothesis
Coming soonResults
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