Vector space/Finite dimensional/Change of basis/Fact

Let $K$ be a field, and let $V$ be a $K$-vector space of dimension $n$. Let $𝔳=v_1,\dots ,v_n$ and $𝔴=w_1,\dots ,w_n$ denote bases of $V$. Suppose that

$v_j=\sum _{i=1}^n{c}_{ij}{w}_i$

with coefficients $c_{ij}\in K$, which we collect into the $n\times n$-matrix

$M_{𝔴}^{𝔳}={\left(c_{ij}\right)}_{ij}.$

Then a vector $u$, which has the coordinates $\left(\begin{array}{c}{s}_1\\ ⋮\\ s_n\end{array}\right)$ with respect to the basis $𝔳$, has the coordinates

$\left(\begin{array}{c}{t}_1\\ ⋮\\ t_n\end{array}\right)=M_{𝔴}^{𝔳}\left(\begin{array}{c}{s}_1\\ ⋮\\ s_n\end{array}\right)=\left(\begin{array}{cccc}{c}_{11}& c_{12}& \dots & c_{1n}\\ c_{21}& c_{22}& \dots & c_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ c_{n1}& c_{n2}& \dots & c_{nn}\end{array}\right)\left(\begin{array}{c}{s}_1\\ ⋮\\ s_n\end{array}\right)$

with respect to the basis $𝔴$.

Proof, Write another proof