how to find transformation matrix of polynomials

Question

I am a student doing my homework.
Let $B1 = (1, (t−1), (t−1)^2, (t−1)^3)$ and $B2 = (1, (t+1), (t+1)^2, (t+1)^3)$. How to find a transformation matrix from $B1$ to $B2$?
I don't really need an exact answer, but the idea how to solve this kind of problems. Also literature or online resources with good explanation are also appreciated.

Answer 1

Hint:

You need to be able to write one basis as a linear combination of the other.

Normally you build up by degree. But here I shall demonstrate for degree 2.

For example $$(t+1)^2=t^2+2t+1 \\ (t-1)^2=t^2-2t+1 \\ \implies (t+1)^2=(t-1)^2+4t $$

Now repeat and try to find $4t$ as a linear combination of the $(t-1)^i$s.