Eigenvalues and Eigenvectors

**test2k6** · May 18th 2007, 04:01 AM

Find a basis B for the domain of T such that the matix of T relative to B is diagonal.
given..
T: P1 -> P1
T(a + bx) = a + (a + 2b)x

I got confused with this hw problem. Thank you.

**Rebesques** · June 16th 2007, 02:48 PM

Identify the polynomial $a+bx$ with the vector $(a \ b)^t$ (transposed, can't bother texing marices

). Then T acts like multiplication by the matrix $T=<br /> (T_1 \ T_2), \ T_1=(1 \ 1)^t , \ T_2=(0 \ 2)^t$ .

How to diagonalize this? Solve $det(T-\lambda I)=0$ for $\lambda$ to get the eigenvalues 1 and 2. The corresponding eigenspaces come from $V(1)=\{a+bx: T(a+bx)=a+bx\}$ and $V(2)=\{a+bx: T(a+bx)=2(a+bx)\}$ . Gladly, each space has one unit eigenvector, and they are orthogonal by theory. Arrange these in a matrix P, and you are done.

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