Page 375 - 35Linear Algebra
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G.2 Systems of Linear Equations                                                               375


                   Hint for Review Question 5

                   The first part for Review Question 5 is simple--just write out the associated
                   linear system and you will find the equation 0 = 6 which is inconsistent.
                   Therefore we learn that we must avoid a row of zeros preceding a non-vanishing
                   entry after the vertical bar.
                      Turning to the system of equations, we first write out the augmented matrix
                   and then perform two row operations
                                                                        
                                                       1  −3    0     6
                                                     1    0    3    −3 
                                                       2   k   3 − k  1
                                                                           
                                                       1   −3     0     6
                                       R 2 −R 1 ;R 3 −2R 1   0
                                            ∼               3     3     −9   .
                                                       0  k + 6  3 − k  −11
                   Next we would like to subtract some amount of R 2 from R 3 to achieve a zero in
                   the third entry of the second column. But if
                                                                   3
                                              k + 6 = 3 − k ⇒ k = − ,
                                                                   2
                   this would produce zeros in the third row before the vertical line. You should
                   also check that this does not make the whole third line zero. You now have
                   enough information to write a complete solution.

                   Planes

                   Here we want to describe the mathematics of planes in space. The video is
                   summarised by the following picture:



















                                            2
                   A plane is often called R because it is spanned by two coordinates, and space
                              3
                   is called R and has three coordinates, usually called (x, y, z). The equation
                   for a plane is
                                                  ax + by + cz = d .


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