Page 342 - 35Linear Algebra
P. 342

342                                                                          Sample Final Exam


                                   (q) The orthogonal complement to a subspace of a vector space.

                               2. Kirchoff’s laws: Electrical circuits are easy to analyze using systems of equa-
                                  tions. The change in voltage (measured in Volts) around any loop due to

                                  batteries | and resistors /\/\/\/\ (given by the product of the current mea-

                                  sured in Amps and resistance measured in Ohms) equals zero. Also, the sum
                                  of currents entering any junction vanishes. Consider the circuit

                                                    1 Ohm                  2 Ohms

                                           I Amps                  13 Amps                 J Amps

                                             60 Volts                80 Volts                V Volts



                                                    3 Ohms                 3 Ohms


                                  Find all possible equations for the unknowns I, J and V and then solve for
                                  I, J and V . Give your answers with correct units.

                               3. Suppose M is the matrix of a linear transformation

                                                                L : U → V

                                  and the vector spaces U and V have dimensions

                                                        dim U = n ,     dim V = m ,

                                  and
                                                                  m 6= n .

                                  Also assume
                                                               kerL = {0 U } .

                                   (a) How many rows does M have?

                                   (b) How many columns does M have?
                                   (c) Are the columns of M linearly independent?
                                                             T
                                   (d) What size matrix is M M?
                                                               T
                                   (e) What size matrix is MM ?
                                           T
                                   (f) Is M M invertible?
                                           T
                                   (g) is M M symmetric?

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