Thread: Find a basis β for V such that [T]β is a diagonal matrix.

1. Find a basis β for V such that [T]β is a diagonal matrix.

This question is sort of related to my other one about diagonalizability. I actually learned something doing that previous question. Maybe someone can give me an explanation or a hint at first.

For each of the following linear operators T on a vector space V, test T for diagonalizability, and if T is diagonalizable, find a basis β for V such that [T]β is a diagonal matrix.

V = P2(R) and T is defined by T(ax2 + bx + c) = cx2 + bx + a.

I'm going to read some of section 5.2 about diagonalizability in my book but the book is abstract and hard to understand. I don't remember what I read about diagonalizability and need to go back a step.

2. We have:

$\displaystyle \begin{Bmatrix}T(1)=x^2\\T(x)=x\\T(x^2)=1\end{matr ix}$

So, the matrix of $\displaystyle T$ with respecto to the canonical basis $\displaystyle B=\{1,x,x^2\}$ is:

$\displaystyle A=\begin{bmatrix}{0}&{0}&{1}\\{0}&{1}&{0}\\{1}&{0} &{0}\end{bmatrix}$

Now, find eigenvalues, eigenvectors, etc.

Fernando Revilla

3. A linear transformation, from vector space V to V, is "diagonalizable" if and only if it has a "complete set of eigenvectors"- that is, a set of eigenvectors that form a basis for the vector space. Writing the linear transformation as a matrix using that basis gives a diagonal matrix.

To write a linear transformation as a matrix using a given basis, apply the linear transformation to each basis vector in turn, then write the result as a linear combination of the basis vectors. The coefficients of those combinations are the colummns of the matrix.

So the first thing you need to do is find the eigenvalues and eigenvectors for this linear transformation.