What does the basis of a linear transformation have to do with the eigenvalues? Larson Edwards Falvo - Elementary Linear Algebra



In exercise 69, what does basis have to do with anything? I think earlier in the text it was shown illustrated that the eigenvalues are the same regardless of the basis. Am I wrong?





 A: You are right. Eigen values do not change with the choice of the basis. I believe the standard basis is specified so that the reader can recognize the vector space in which the problem is set in. Also, standard basis makes the calculations bit easier.
A: The eigenvalues and eigenvectors of a linear transformation are intrinsic, that is can be defined without referring to a basis of the vector space. But to compute the eigenvalues, the matrix of the linear transformation wrt some basis is helpful.
To give a rough analogue look at the concept of prime numbers and factorization. This is intrinsic; every number is uniquely expressible as a product of powers of prime numbers. However if the number is presented in base 10 form such as, for example, 27475, it is immediately clear  that it must be a multiple of 25, so we can divide by 25 get a smaller number and  discover other factors.
A: This is not an answer, but is a pain to write  as a comment.
The question is ambiguously phrased.
Solving the equation $T(a_0+a_1 x + a_2 x^2) = \lambda ( a_0+a_1 x + a_2 x^2 )$
for $\lambda$ and $(a_0,a_1,a_2) \neq (0,0,0)$ is exactly equivalent to solving
$\begin{bmatrix} 0 & -3 & 5 \\
-4 & 4 & -10 \\
0 & 0 & 4 \end{bmatrix} \begin{bmatrix} a_0 \\ a_1 \\ a_2 \end{bmatrix} = \lambda \begin{bmatrix} a_0 \\ a_1 \\ a_2 \end{bmatrix}  $.
Then $a_0+a_1 x + a_2 x^2$ is an eigenvector of $T$ corresponding to the
eigenvalue $\lambda$ and $\begin{bmatrix} a_0 \\ a_1 \\ a_2 \end{bmatrix}$ is an eigenvector of the matrix of $T$ with basis $\{1,x,x^2\}$ corresponding to the
eigenvalue $\lambda$.
It is not clear to me whether they are looking for the answer $a_0+a_1 x + a_2 x^2$ or $\begin{bmatrix} a_0 \\ a_1 \\ a_2 \end{bmatrix}$ (with appropriate numbers, of course).
