mat67-Homework7_Solutions

mat67-Homework7_Solutions - MAT067 University of California...

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MAT067 University of California, Davis Winter 2007 Solutions to Homework Set 7 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 (1) cx 1 + dx 2 . (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. Solution : Assume that ad bc 6 = 0 and consider the combination d (1) b (2), which can be written as ( ad bc ) x 1 = 0. Since by assumption the factor ( ad bc ) doesn’t vanish, we must conclude that x 1 . Now consider the combination c (1) + a (2). This similarly yields ( ad bc ) x 2 =0, so, again using the assumption, we must conclude that x 2 as well. Therefore, assuming ad bc 6 ,wehavefoundtha t x 1 = x 2 = 0 is the unique solution. If on the other hand, ad bc ,then x 1 = d, x 2 = c and x 1 = b, x 2 = a are solutions. If at least one of a, b, c, d is nonzero, this yields a non-trivial solution. If a = b = c = d = 0, then certainly all x 1 ,x 2 F satisfy (1) and (2). 2. Let A = ± ab cd ² F 2 × 2 , and recall that we can deFne 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.
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2 Solution : One method for Fnding the eigen-information of T is to analyze the solutions of the matrix equation Av = λv for λ F and v F 2 . In particular, using the deFnition of eigenvector and eigenvalue, v is an eigenvector associated to the eigenvalue λ if and only if Av = T ( v )= λv . A simpler method involves the equivalent matrix equation ( A λI ) v =0 ,where I denotes the identity map on F 2 . In particular, 0 6 = v F 2 is an eigenvector for T associated to the eigenvalue λ F if and only if the system of linear equations ( a λ ) v 1 + bv 2 cv 1 +( d λ ) v 2 ± (3) has a non-trivial solution. Moreover, the system of equations (3) has a non-trivial solution if and only if the polynomial p ( λ )=( a λ )( d λ ) bc evaluates to zero (see Problem 1). In other words, the eigenvalues for T are exactly the λ F for which p ( λ )=0 , and the eigenvectors for T associated to an eigenvalue λ are exactly the non-zero vectors v = ² v 1 v 2 ³ F 2 that satisfy the system of equations (3). 3. ±ind eigenvalues and associated eigenvectors for the linear operators on F 2 deFned by each given 2 × 2 matrix.
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mat67-Homework7_Solutions - MAT067 University of California...

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