For the following linear operator T on a vector space V, test T for diagonalizability,?

and if T is diagonalizable, find a basis beta for V such that [T]beta is a diagonal matrix.


V=P2(R) and T is defined by T(ax^2+ bx+ c)= cx^2+ bx+a.

For the following linear operator T on a vector space V, test T for diagonalizability,?
The representation of T relative to the basis {1,x,x²} is





T = [0 0 1; 0 1 0; 1 0 0] (the semicolons separate lines)





This matrix is diagonalizable, as you can see, for example, from its characteristic polynomial:





p(λ) = λ³ - λ² - λ + 1





Whose roots are 1 (with multiplicity 2) and -1, and the corresponding eigenspaces are:





{(0,1,0),(1,0,1)} and {(-1,0,1)}





These span V, and the matrix representing T relative to this basis is diagonal.

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