Page 333 - 35Linear Algebra
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                      6. Suppose λ is an eigenvalue of the matrix M with associated eigenvector v.
                         Is v an eigenvector of M k  (where k is any positive integer)? If so, what
                         would the associated eigenvalue be?
                         Now suppose that the matrix N is nilpotent, i.e.
                                                           k
                                                         N = 0
                         for some integer k ≥ 2. Show that 0 is the only eigenvalue of N.
                                         !
                                    3 −5
                                                         12
                      7. Let M =           . Compute M . (Hint: 2   12  = 4096.)
                                    1 −3
                      8. The Cayley Hamilton Theorem: Calculate the characteristic polynomial

                                                     a  b
                         P M (λ) of the matrix M =         . Now compute the matrix polynomial
                                                     c d
                         P M (M). What do you observe? Now suppose the n×n matrix A is “similar”
                         to a diagonal matrix D, in other words

                                                       A = P  −1 DP

                         for some invertible matrix P and D is a matrix with values λ 1 , λ 2 , . . . λ n
                         along its diagonal. Show that the two matrix polynomials P A (A) and P A (D)
                         are similar (i.e. P A (A) = P −1 P A (D)P). Finally, compute P A (D), what can
                         you say about P A (A)?

                      9. Define what it means for a set U to be a subspace of a vector space V .
                         Now let U and W be non-trivial subspaces of V . Are the following also
                         subspaces? (Remember that ∪ means “union” and ∩ means “intersection”.)

                          (a) U ∪ W
                          (b) U ∩ W
                                                       3
                         In each case draw examples in R that justify your answers. If you answered
                         “yes” to either part also give a general explanation why this is the case.

                     10. Define what it means for a set of vectors {v 1 , v 2 , . . . , v n } to (i) be linearly
                         independent, (ii) span a vector space V and (iii) be a basis for a vector
                         space V .
                                                          3
                         Consider the following vectors in R
                                                                            
                                           −1               4                 10
                                                            5
                                     u =   −4   ,   v =     ,    w =       7   .
                                             3              0               h + 3
                                                                     3
                         For which values of h is {u, v, w} a basis for R ?

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