Define

by

. Show that

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

it is

, am i supposed to use this? If so how?

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