1. ## Diagonalizable Transformation

Define $T: R[X]_{\leq 2} \rightarrow R[X]_{\leq 2}$ by $(Tf)(t)=(1-t^2)f''(t)-tf'(t) + 2f(t)$. Show that $T$is diagonalizable and find a basis of eigenvectors.

Now I know how to show a matrix is diagonalizable and find its eigenvectors. My problem is im not sure how to get the matrix of this transformation. I know generally you just plug the elements of the standard basis into the transformation one by one to get the matrix of the transformation in the standard basis. However im not sure what the standard basis is in this case? for $R[X]_{\leq 2}$ it is $1,x,x^2$, am i supposed to use this? If so how?

Any help finding the matrix of this transformation would be great.

2. Originally Posted by joe909
Define $T: R[X]_{\leq 2} \rightarrow R[X]_{\leq 2}$ by $(Tf)(t)=(1-t^2)f''(t)-tf'(t) + 2f(t)$. Show that $T$is diagonalizable and find a basis of eigenvectors.

Now I know how to show a matrix is diagonalizable and find its eigenvectors. My problem is im not sure how to get the matrix of this transformation. I know generally you just plug the elements of the standard basis into the transformation one by one to get the matrix of the transformation in the standard basis. However im not sure what the standard basis is in this case? for $R[X]_{\leq 2}$ it is $1,x,x^2$, am i supposed to use this? If so how?

Any help finding the matrix of this transformation would be great.

Who cares what basis you choose to represent your transformation in matrix form? After all we know ANY of them will be diagonalizable if one of them is, or none will be if one of them isn't.

So you can choose as well the basis $\{1,x,x^2\}$ :

$T(1)=2=2\cdot 1 + 0\cdot x + 0\cdot x^2$

$T(x)=-x+2x = x = 0\cdot 1 + 1\cdot x + 0\cdot x^2$

And etc. When you have this matrix, say $A$ , evaluate its characteristic polymomial $p_A(x):=\det(xI-A)$ to get its eigenvalues and etc.

Tonio