coordinate vector

$[\mathbf x]_{\mathcal B}$ — the unique list of coefficients expressing $\mathbf x$ in a given basis.

Given a basis $\mathcal B=\{\mathbf b_1,\dots,\mathbf b_n\}$ , every $\mathbf x\in V$ has a unique representation $\mathbf x=c_1\mathbf b_1+\cdots+c_n\mathbf b_n$ ; the resulting list $[\mathbf x]_{\mathcal B} = (c_1,\dots,c_n)$ is the coordinate vector of $\mathbf x$ relative to $\mathcal B$ . It always lives in $\mathbb R^n$ regardless of what $V$ 's elements actually look like, and finding it means solving $M\mathbf c=\mathbf x$ where $M$ 's columns are the basis vectors.