So, it is clear that $A = (1, x, x^2, x^3)$ and $B = (1, x + 1, (x+1)^2, (x+1)^3)$ each in their own right form a basis of the $\mathbb{R}$-vector space of polynomials of at degree at most $3$. My question is, what is the change of basis matrix from $A$ to $B$? Could someone work this out in a detailed manner so a beginner like me can follow?
After some algebra, I ended up getting$$\text{Stuff} = \begin{pmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 1\end{pmatrix}.$$And checking to see if this is correct, my $1$ in $A$ corresponds to the vector$$\begin{pmatrix} 1 \\ 0 \\ 0 \\ 0\end{pmatrix}.$$And$$(\text{Stuff})\begin{pmatrix} 1 \\ 0 \\ 0 \\ 0\end{pmatrix} = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0\end{pmatrix},$$which corresponds to the element $1$ in the basis $B$.
Let's also check $x^3$ in $A$, which is a bit different. This corresponds to$$\begin{pmatrix}0 \\ 0 \\ 0 \\ 1\end{pmatrix}.$$And$$(\text{Stuff})\begin{pmatrix} 0 \\ 0 \\ 0 \\ 1\end{pmatrix} = \begin{pmatrix} 1 \\ 3 \\ 3 \\ 1\end{pmatrix},$$which corresponds to$$x^3 + 3x^2 + 3x + 1 = (x+1)^3$$in the basis $B$.
The matrix
$$\text{Stuff} = \begin{pmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 1\end{pmatrix}$$
which has for columns the components of the basis vectors of $B$ with respect to the basis vectors of $A$, represents the change of basis form basis $B$ to basis $A$.
Therefore $(\text{Stuff})^{-1}$ represents the change of basis form basis $A$ to basis $B$.
Refer also to the related