Question 7.4.5: Consider the linear transformation T f (x) = f (2x − 1) from......

Here it would be hard to find eigenvalues and eigenfunctions “by inspection”; we need a systematic approach. The idea is to find the matrix A of T with respect to some convenient basis 𝔄. Then we can determine whether A is diagonalizable, and, if so, we can find an eigenbasis for A. Finally we can transform this basis back into P_2 to find an eigenbasis 𝔅 for T .

We will use a commutative diagram to find the matrix A of T with respect to the standard basis 𝔄 = (1, x, x^2).

The upper triangular matrix A has the three distinct eigenvalues, 1, 2, and 4, so that A is diagonalizable, by Theorem 7.3.4. A straightforward computation produces the eigenbasis

\begin{bmatrix}1\\0\\0\end{bmatrix}, \begin{bmatrix}-1\\1\\0\end{bmatrix}, \begin{bmatrix}1\\-2\\1\end{bmatrix}

for A. Transforming these vectors back into P_2, we find the eigenbasis 𝔅 for T consisting of

1, \ x-1, x^2 − 2x + 1 = (x − 1)^2.

To check our work, we can verify that these are indeed eigenfunctions of T :

T (1) = 1
T (x − 1) = (2x − 1) − 1 = 2x − 2 = 2(x − 1)
T((x − 1)^2)= ((2x − 1) − 1)^2 = (2x − 2)^2 = 4(x − 1)^2.  √

The 𝔅-matrix of T is

B=\begin{bmatrix}1&0&0\\0&2&0\\0&0&4\end{bmatrix}.

Consider Figure 2, where

S=\begin{bmatrix}1&-1&1\\0&1&-2\\0&0&1\end{bmatrix}

is the change of basis matrix from 𝔄 to 𝔅.