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## Homework Statement

Define T: R2-->R2 by T(x)=Ax

Find a basis B for R2 with the property that [T]_B is diagonal.

A=

0 1

-3 4

## The Attempt at a Solution

The eigenvalues of a diagonal matrix are its diagonal entries, so here the eigenvalues are 1, and -3. For eigenvalue=1 I get the basis [1,1] and for eigenvalue -3 I do not get a linearly dependent system. However, accidentally solved the system for eigenvalue=3 and got the basis [1,3] which is supposed to be correct. Can anyone see what I'm doing wrong?