For the question below, I am having trouble writing out the matrix for $T^k$. I am not sure how I could write the matrix for $T$ in this case. What I believe is troubling me about this question is how $T$ is defined. How should I interpret the composition of $T$ with itself $k$ times?
Thanks in advance
$P_3$ refers to the polynomial with degree up to $3$, and $p'$ refers to the first derivative of $p$
Let $T:P_3(\mathbb{R})\rightarrow P_3(\mathbb{R})$ be the operator $$T:p\rightarrow p'-p''+p'''$$
Write the matrix for $T^k=T\circ\cdot\cdot\cdot\circ T$
(with $k$ factors) $\forall k=1,2,...,$ using the standard basis of $P_3$
The vectors in the standard base (in the polynomial representation) are $(1,x,x^2,x^3)$.
Now we need to find the image of the elements under the operator: $$ 1 \rightarrow 0$$ $$ x \rightarrow 1$$ $$x^2\rightarrow -2+2x$$ $$x^3\rightarrow 6-6x+3x^2$$.
Now we can project those images on the base obtaining the matrix representation of the operator: $$[T]= \begin{bmatrix} 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0\\ -2 & 2 & 0 & 0\\ 6 & -6 & 3 & 0 \end{bmatrix} $$.
Now you just need to raise this matrix to the $k$th power.