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.

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- May 18th 2007, 04:01 AMtest2k6Eigenvalues and Eigenvectors
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. - Jun 16th 2007, 02:48 PMRebesques
Identify the polynomial $\displaystyle a+bx$ with the vector $\displaystyle (a \ b)^t$ (transposed, can't bother texing marices :o ). Then T acts like multiplication by the matrix $\displaystyle T=

(T_1 \ T_2), \ T_1=(1 \ 1)^t , \ T_2=(0 \ 2)^t $.

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