CorollaryProperties of coordinate change

Given ordered bases \(\OrdVSpcBss{B}\), \(\OrdVSpcBss{C}\), and \(\OrdVSpcBss{D}\) of a subvector space \(\VSpc{W}\) of \(\RNrSpc{n}\), the associated coordinate conversion matrices are related by

\(\CoordTrafoMtrx{C}{D}{B}\)\(=\)\(\CoordTrafoMtrx{C}{D}{C} \CoordTrafoMtrx{C}{C}{B}\)

Proof

For an arbitrary vector \(\Vect{x}\) in \(\VSpc{W}\) we find

\[\CoordVect{x}{D} = \CoordTrafoMtrx{C}{D}{C}\CoordVect{x}{C} = \CoordTrafoMtrx{C}{D}{C}\left( \CoordTrafoMtrx{C}{C}{B} \CoordVect{x}{B}\right)\]

Therefore the product matrix \(\left( \CoordTrafoMtrx{C}{D}{C} \CoordTrafoMtrx{C}{C}{B}\right)\) converts from \(\OrdVSpcBss{B}\)-coordinates to \(\OrdVSpcBss{D}\)-coordinates. But we know that there is only one matrix with this property, namely \(\CoordTrafoMtrx{C}{D}{B}\). Therefore these two matrices are equal, and this proves the corollary.