Dual mapping/Dual basis/Matrix/Fact

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

${\displaystyle \varphi \colon V\longrightarrow W}$

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

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

with respect to the given bases.

Then the

dual mapping

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

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

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