Consider the space of polynomials of degree at most4,P4. The standard basis forP4is given by the monomials:

E={1,x,x2,x3,x4}. However, there are other bases for the polynomials which may be more convenient to work with in some settings; two are detailed below.

The Chebyshev polynomials are defined by the conditionsU0(x)=1,U1(x)=2x, and forn>1,Un+1(x)=2xUn(x)Un1(x). In particular, the first five Chebyshev polynomials are as follows:

B={1,2x,4x21,8x34x,16x412x2+1}.

Their general form is not as easily presented, but the first five Hermite polynomials are the following:

C={1,x,x21,x33x,x46x2+3}.

Compute the following change of basis matrices.

A.[I]EB, the change of basis matrix fromB-coordinates toE-coordinates.

B.[I]EC, the change of basis matrix fromC-coordinates toE-coordinates.

C.[I]BC, the change of basis matrix fromC-coordinates toB-coordinates.