Consider the linear transformation l p 3 7 p 3

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4. (20 points) Consider the linear transformation L : P 3 7→ P 3 defined by L ( p ) = (1 - x 2 ) d 2 p dx 2 - 2 x dp dx a) Find the matrix representation of L in the standard basis { 1 , x, x 2 , x 3 } b) Is L onto? one-to-one? c) Find the eigenvalues and eigenvectors of L . Your eigenvectors must be ele- ments of P 3 . d) Give a basis B of P 3 such that [ L ] B is diagonal. In your basis what is [ L ] B ? Solution.
c) Since the matrix is triangular the eigenvalues are the diagonal entries λ = 0 , - 2 , - 6 , - 12. We’ve actually already calculated the first two eigenvectors as well. Since L (1) = 0 = 0 · 1, and L ( x ) = - 2 x = - 2 · x we have that v 0 ( x ) = 1 and v - 2 ( x ) = x are the eigenvectors for λ = 0 and λ = - 2 respectively. To compute the other eigenvectors we proceed as usual: [ L ] S + 6 I = 6 0 2 0 0 4 0 6 0 0 0 0 0 0 0 - 6 3 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 v - 6 = 1 0 - 3 0 v - 6 ( x ) = 1 - 3 x 2 [ L ] S + 12 I = 12 0 2 0 0 10 0 6 0 0 6 0 0 0 0 0 1 0 0 0 0 5 0 3 0 0 1 0 0 0 0 0 v - 12 = 0 3 0 - 5 v - 12 ( x ) = 3 x - 5 x 3 d) We have now a basis of P 3 consisting of eigenvectors of L : B = { 1 , x, 1 - 3 x 2 , 3 x - 5 x 3 } . In this basis the matrix representation of L is diagonal: L B = 0 0 0 0 0 - 2 0 0 0 0 - 6 0 0 0 0 - 12 .
5. (15 points) For each of the following statements, decide whether it is true or false and give a brief proof or an explicit counterexample. a) If A, B , and P are n × n matrices which satisfy the equation AP = PB , then A and B are similar.
b) If A and B are n × n matrices and 0 is an eigenvalue of A , then 0 is an eigenvalue of AB .

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