Page 322 - 35Linear Algebra
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322                                                                       Sample First Midterm


                                   (d) What are the vectors X 0 and Y i called and which matrix equations do
                                       they solve?
                                   (e) Check separately that X 0 and each Y i solve the matrix systems you
                                       claimed they solved in part (d).

                               3. Use row operations to invert the matrix

                                                                           
                                                               1   2   3   4
                                                               2   4   7 11
                                                                           
                                                                           
                                                               3   7 14 25
                                                                           
                                                               4 11 25 50

                                              2   1                 T   −1
                               4. Let M =             . Calculate M M     . Is M symmetric? What is the
                                              3 −1
                                                                               2
                                  trace of the transpose of f(M), where f(x) = x − 1?
                               5. In this problem M is the matrix


                                                                    cos θ  sin θ
                                                           M =
                                                                  − sin θ cos θ
                                  and X is the vector

                                                                       x
                                                                X =       .
                                                                       y
                                  Calculate all possible dot products between the vectors X and MX. Com-
                                  pute the lengths of X and MX. What is the angle between the vectors MX
                                  and X. Draw a picture of these vectors in the plane. For what values of θ
                                  do you expect equality in the triangle and Cauchy–Schwartz inequalities?

                               6. Let M be the matrix
                                                             1 0 0 1 0 0
                                                                             
                                                             0 1 0 0 1 0
                                                                             
                                                                             
                                                             0 0 1 0 0 1
                                                                             
                                                                             
                                                            0 0 0 1 0 0      
                                                                             
                                                            0 0 0 0 1 0      
                                                             0 0 0 0 0 1
                                                       k
                                  Find a formula for M for any positive integer power k. Try some simple
                                  examples like k = 2, 3 if confused.
                               7. What does it mean for a function to be linear? Check that integration is a
                                  linear function from V to V , where V = {f : R → R | f is integrable} is a
                                  vector space over R with usual addition and scalar multiplication.


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