Dual mapping/Dual basis/Matrix/Fact

Let ${}K$ be a field, let ${}V$ denote an ${}n$-dimensional ${}K$-vector space with a basis ${}{\mathfrak {v}}=v_{1},\ldots ,v_{n}$, and let ${}W$ be an ${}m$-dimensional vector space with a basis ${}{\mathfrak {w}}=w_{1},\ldots ,w_{m}$. Let ${}v_{1}^{*},\ldots ,v_{n}^{*}$ and ${}w_{1}^{*},\ldots ,w_{m}^{*}$ be the corresponding dual bases. Let

$\varphi \colon V\longrightarrow W$

be a linear mapping, and suppose that it is described by the ${}m\times n$-matrix

${}M=M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi )=(a_{ij})_{ij}\,$

with respect to the given bases.

Then the

dual mapping

$\varphi ^{*}\colon {W}^{*}\longrightarrow {V}^{*}$

is described by the transposed matrix ${}{{\left(M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi )\right)}^{\text{tr}}}$ with respect to the dual bases of

${}{W}^{*}$ and ${}{V}^{*}$.

Proof, Write another proof