Find a basis B for the domain of T such that the matix of T relative to B is diagonal.
T: P1 -> P1
T(a + bx) = a + (a + 2b)x
I got confused with this hw problem. Thank you.
Identify the polynomial with the vector (transposed, can't bother texing marices ). Then T acts like multiplication by the matrix .
How to diagonalize this? Solve for to get the eigenvalues 1 and 2. The corresponding eigenspaces come from and . Gladly, each space has one unit eigenvector, and they are orthogonal by theory. Arrange these in a matrix P, and you are done.