Let V {\displaystyle {}V} be a finite-dimensional K {\displaystyle {}K} -vector space. Let v 1 , … , v n {\displaystyle {}v_{1},\ldots ,v_{n}} be a basis of V {\displaystyle {}V} with the dual basis v 1 ∗ , … , v n ∗ {\displaystyle {}v_{1}^{*},\ldots ,v_{n}^{*}} , and let w 1 , … , w n {\displaystyle {}w_{1},\ldots ,w_{n}} be another basis with the dual basis w 1 ∗ , … , w n ∗ {\displaystyle {}w_{1}^{*},\ldots ,w_{n}^{*}} , and with
where ( b i j ) i j = ( A − 1 ) tr {\displaystyle {}{\left(b_{ij}\right)}_{ij}={{\left(A^{-1}\right)}^{\text{tr}}}} is the transposed matrix of the inverse matrix of
be a
finite-dimensional
-vector space.
Let 
be a
basis
of with the
dual basis
, and let 
be another basis with the
dual basis 
, and with
is the
transposed matrix
of the
inverse matrix
of
