On page 69, “Linear Algebra Done Wrong”, says:
8.1. Coordinate vector. Let $V$ be a vector space with a basis $B:=\left\{b_{1},\ldots ,b_{n}\right\}$. Any vector $v \in V$ admits a unique representation as a linear combination $$v=x_{1}b_{1}+\ldots +x_{n}b_{n}=\sum ^{n}_{k=1}x_{k}b_{k}$$
The numbers $x_{1},\ldots ,x_{n}$ are called the coordinates of the vector $v$ in the basis B. It is convenient to join these coordinates into the so-called coordinate vector of v relative to the basis B, which is the column vector $$[v]_B:=\begin{pmatrix} x_{1} \\ \vdots \\ x_{n} \end{pmatrix}\in \mathbb{F}^{n}$$ Note that the mapping $$v\to[v]_B$$ is an isomorphism between $V$ and $\mathbb{F}^n$. It transforms the basis $b_{1},\ldots ,b_{n}$ to the standard basis $e_{1},\ldots ,e_{n}$.
I do not understand the last paragraph. I thought the output of the coordinate function gave the coordinates with respect to basis $B$. Those are precisely the coefficients that multiply each of basis $B$ vectors. Can someone throw some light?