The matrix [` T`(

From the red arrows in the diagram, we conclude the following (the second equality follows from this formula).

For a linear transformation ** T**, the matrices with regard to different bases are related by

[` T`(

Example For the linear transformation and bases in this example, we found the matrix for the transformation.

[(THermite)] = [_{C} |
1 | 0 | -1 | 0 | ] |

1 | 1 | 0 | -2 | ||

0 | 1 | 0 | -3 | ||

0 | 1 | 0 | -3 |

Now we change the basis `B` to `Chebyshev` = {1, `t`, 2`t`^{2}, 4`t`^{3} - 3`t`} and keep the basis `C` as before. To find the new matrix [` T`(

[Chebychev] = [_{Hermite} |
1 | 0 | 2 | 0 | ] |

0 | 1 | 0 | 9 | ||

0 | 0 | 2 | 0 | ||

0 | 0 | 0 | 4 |

Thus we have

[(TChebychev)] = [_{C}(THermite)] [_{C}Chebychev] = [_{Hermite} |
1 | 0 | 0 | 0 | ]. |

1 | 1 | 2 | 1 | ||

0 | 1 | 0 | -3 | ||

0 | 1 | 0 | -3 |

We may compare the result with the direct application of ** T** to

(1) = [T |
1 | 1 | ], (Tt) = [ |
0 | 1 | ], (2Tt^{2}) = [ |
0 | 2 | ], (4Tt^{3} - 3t) = [ |
0 | 1 | ] |

0 | 0 | 1 | 1 | 0 | 0 | -3 | -3 |