### Change of Basis

##### 4. Change of the matrix for linear transformation

The matrix [T(B)]C of a linear transformation T: VW depends on the choices of bases B and C. If B' and C' are other choices, then everything fits into a diagram (see this formula). 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(B')]C' = [C]C' [T(B)]C [B]B'-1 = [C]C' [T(B)]C [B']B.

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

 [T(Hermite)]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, 2t2, 4t3 - 3t} and keep the basis C as before. To find the new matrix [T(Chebyshev)]C, we recall the matrix for the change of coordinates from another example.

 [Chebychev]Hermite = [ 1 0 2 0 ] 0 1 0 9 0 0 2 0 0 0 0 4

Thus we have

 [T(Chebychev)]C = [T(Hermite)]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 Chebychev.

 T(1) = [ 1 1 ], T(t) = [ 0 1 ], T(2t2) = [ 0 2 ], T(4t3 - 3t) = [ 0 1 ] 0 0 1 1 0 0 -3 -3