mat67-Homework7

# mat67-Homework7 - MAT067 University of California Davis...

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MAT067 University of California, Davis Winter 2007 Homework Set 7: More Exercises on Eigenvalues Directions : Please work on all of the following exercises. Submit your solutions to Problems 3(d) and 5(b) as your Calculational Problems and Problems 1 and 2 as your Proof-Writing Problems at the beginning of lecture on February 23, 2007. As usual, we are using F to denote either R or C . 1. Let a, b, c, d F and consider the system of equations given by ax 1 + bx 2 =0 cx 1 + dx 2 . Note that x 1 = x 2 = 0 is a solution for any choice of a, b, c ,and d . Prove that this system of equations has a non-trivial solution if and only if ad bc =0. 2. Let A = ± ab cd ² F 2 × 2 , and recall that we can de±ne a linear operator T ∈L ( F 2 )on F 2 by setting T ( v )= Av for each v = ± v 1 v 2 ² F 2 . Show that the eigenvalues for T are exactly the λ F for which p ( λ )=0 ,wh e r e p ( z )=( a z )( d z ) bc . Hint : Write the eigenvalue equation Av = λv as ( A λI ) v = 0 and use Problem 1. 3. Find eigenvalues and associated eigenvectors for the linear operators on F 2 de±ned by the following 2 × 2 matrices:

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mat67-Homework7 - MAT067 University of California Davis...

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