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Consider the space of polynomials of degree at most $4$, $\mathbb{P}_4$. The standard basis for $\mathbb{P}_4$ is given by the monomials: $\mathcal{E} = \{1,x,x^2,x^3,x^4\}$. 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 (of the second type) are useful in approximation theory and polynomial interpolation, and are defined by the conditions $U_0(x) = 1$, $U_1(x) = 2x$, and for $n > 1$, $U_{n+1}(x) = 2xU_n(x) - U_{n-1}(x)$. In particular, the first five Chebyshev polynomials are as follows: $$\mathcal{B} = \{1, 2x, 4x^2-1, 8x^3-4x, 16x^4-12x^2+1\}.$$
The Hermite polynomials likewise have many uses, being related to the behaviour of a quantum harmonic oscillator and random matrix theory. Their general form is not as easily presented, but the first five Hermite polynomials are the following: $$\mathcal{C} = \{1, x, x^2-1, x^3-3x, x^4-6x^2+3\}.$$
Compute the following change of basis matrices.