For a change of basis question, I have the following bases:
$$\epsilon = {1, x^1, x^2, x^3}$$ and $$\beta = {1, (x-1), (x-1)^2, (x-1)^3}$$
I believe that $\epsilon$ can be expressed as $$ \begin{bmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{bmatrix} $$
However, I am unsure on how to proceed with turning $\beta$ into a matrix. I have an attempt that translates to $$ \begin{bmatrix} 1&0&0&0\\ 0&(x-1)&0&0\\ 0&0&(x-1)&0\\ 0&0&0&(x-1) \end{bmatrix} $$ but I highly doubt that is correct.
Edit: I have checked this question here: How do I express ordered bases for polynomials as a matrices? Linear Algebra.
but it lead me nowhere for my specific qualm.
You expand the expressions and check the coefficients and put them in the right place in the matrix.
First column: $$(1)\cdot 1$$ Second column: $$1\cdot x + (-1)\cdot 1$$ Third column: $$(x-1)^2 = 1\cdot x^2 + (-2)\cdot x + 1\cdot 1$$ Fourth column: $$(x-1)^3 = 1\cdot x^3 + (-3)\cdot x^2 + 3\cdot x + (-1) \cdot 1$$
$$\begin{bmatrix}1&-1&1&-1\\0&1&-2&3\\0&0&1&-3\\0&0&0&1\end{bmatrix}$$