Finding the inverse of a linear transformation

Question

Let $A:P_3 \to P_3$ be linear operator such that $$Ap(x)=\int_0^1p(x+t)dt$$ where $p \in P_3$. Find $A(e)$ if $(e)=\{1,x,x^2,x^3\}$ and $A^{-1}(2x-x^3)$

I just started learning about linear operators and I got stuck in this problem. I was able to find $A(e)$ $$ A(e)=\begin{bmatrix} 1 & \frac12 & \frac13 & \frac14 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & \frac32 \\ 0 & 0 & 0 & 1\\ \end{bmatrix}$$

But I don't know how to find the inverse $A^{-1}(2x-x^3)$. I would really appreciate some help.

Answer 1

But I don't know how to find the inverse $A^{-1}(\color{blue}{2x-x^3})$.

Hint

Now that you already have $A(e)$, this comes down to solving: $$\begin{bmatrix} 1 & \frac12 & \frac13 & \frac14 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & \frac32 \\ 0 & 0 & 0 & 1\\ \end{bmatrix} \begin{bmatrix} a \\ b \\ c \\ d \end{bmatrix}=\begin{bmatrix} \color{blue}{ 0 \\ 2 \\ 0 \\ -1} \end{bmatrix}$$