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Consider a linear transformation $L:\mathbb{R[X]_{\leq3}} \rightarrow \mathbb{R[X]_{\leq3}}:f \rightarrow f'' - 4f' + f$. Find the transformation matrix $: T_{\alpha}^{\alpha}$ where $\alpha = ({X,1+X,X+X^2,X^3)}$.

I have no idea how to get that matrix with polynomials.

Need a little bit explaination :( I'm stucked and did not find anywhere any good source.

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The first column will be $L$ of the first basis vector, written as a coordinate vector. So: $$\eqalign{ \langle\hbox{first basis vector}\rangle&=X\cr L(X)&=-4+X\cr \langle\hbox{coordinate vector}\rangle&=(5,-4,0,0)\cr \langle\hbox{matrix}\rangle&=\pmatrix{5&?&?&?\cr-4&?&?&?\cr0&?&?&?\cr0&?&?&?\cr}\cr}$$ I'm sure you can do the rest similarly, using the other basis vectors.

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