If \(\OrdVSpcBss{B}\) and \(\OrdVSpcBss{C}\) are ordered bases of a subvector space \(\VSpc{W}\) of \(\RNrSpc{n}\), then
| \(\CoordTrafoMtrx{C}{B}{C}\) | \(=\) | \(\left(\CoordTrafoMtrx{C}{C}{B}\right)^{-1}\) |
For an arbitrary vector \(\Vect{x}\) in \(\VSpc{W}\) we find
\[\CoordVect{x}{B} = \CoordTrafoMtrx{C}{B}{C} \CoordTrafoMtrx{C}{C}{B} \CoordVect{x}{B} \quad\text{and}\quad \CoordVect{x}{C} = \CoordTrafoMtrx{C}{C}{B} \CoordTrafoMtrx{C}{C}{B} \CoordVect{x}{C}\]Now, if \(\Dimnsn{\VSpc{W}}=r\), we also have that
\[\CoordVect{x}{B} = \IdMtrx{r} \CoordVect{x}{B} \quad\text{and}\quad \CoordVect{x}{C} = \IdMtrx{r} \CoordVect{x}{C}\]As coordinate conversion matrices are unique, this gives
\[\CoordTrafoMtrx{C}{B}{C} \CoordTrafoMtrx{C}{C}{B} = \IdMtrx{n} \quad\text{and}\quad \CoordTrafoMtrx{C}{C}{B} \CoordTrafoMtrx{C}{B}{C} = \IdMtrx{n}\]This implies the corollary on reversing coordinate conversion.