Page 273 - 35Linear Algebra
P. 273

14.7 Review Problems                                                                          273



                          (c) Suppose the vectors a, b and c, d are orthogonal. What can
                              you say about M in this case? (Hint: think about what M        T  is
                              equal to.)

                      2. Suppose S = {v 1 , . . . , v n } is an orthogonal (not orthonormal) basis
                                                                                    i
                              n
                         for R . Then we can write any vector v as v =         P   c v i for some
                                                                                  i
                                    i
                                                                         i
                         constants c . Find a formula for the constants c in terms of v and the
                         vectors in S.
                                                          Hint



                                                                       3
                      3. Let u, v be linearly independent vectors in R , and P = span{u, v} be
                         the plane spanned by u and v.

                                             ⊥
                          (a) Is the vector v := v −  u·v u in the plane P?
                                                      u·u
                                                                           ⊥
                         (b) What is the (cosine of the) angle between v and u?
                                                                                              ⊥
                          (c) How can you find a third vector perpendicular to both u and v ?
                                                                    3
                         (d) Construct an orthonormal basis for R from u and v.
                          (e) Test your abstract formulæ starting with


                                                u = 1, 2, 0 and v = 0, 1, 1 .



                                                          Hint



                                                           4
                      4. Find an orthonormal basis for R which includes (1, 1, 1, 1) using the
                         following procedure:


                          (a) Pick a vector perpendicular to the vector


                                                                
                                                                 1
                                                                 1
                                                                
                                                          v 1 =   
                                                                 1
                                                                
                                                                 1
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