1. Math 307 Spring, 2003 Hentzel Time: 1:10-2:00 MWF Room: 1324 Howe Hall Instructor: Irvin Roy Hentzel Office 432 Carver Phone 515-294-8141 E-mail: hentzel@iastate.edu http://www.math.iastate.edu/hentzel/class.307.ICN Text: Linear Algebra With Applications, Second Edition Otto Bretscher

2. Monday, Apr 7 Chapter 6.3 Page 286 Problems 1 through 44 Main Idea: Learn interesting uses of the determinant. Key Words: Expansion Factor, Cramer's Rule Goal: Learn to appreciate the Determinant's wonderful Qualities.

3. Previous Assignment Friday, April 4 Chapter 6.2 Page 266 Problem 4,14,46,48 Page 266 Problem 4 Find the determinant | 0 2 1 0 1 | | 0 0 2 0 2 | | 0 5 3 9 9 | | 0 7 4 0 1 | | 3 9 5 4 8 |

4. | 2 1 0 1 | 3 | 0 2 0 2 | | 5 3 9 9 | | 7 4 0 1 | | 2 1 1 | 3*9 | 0 2 2 | | 7 4 1 | | 2 1 1 | 3*9 | -4 0 0 | | 7 4 1 |

5. | 1 1 | 3*9*4 | 4 1 | 3*9*4*( - 3) = - 324

6. Page 266 Problem 14 | V1 | Det | V2 | = 8 | V3 | | V4 | | 6 V1 + 2 V4 | | 2(3V1+V4) | Find Det | V2 | = Det | V2 | | V3 | | V3 | | 3 V1 + V4 | | 3 V1 + V4 |

7. | 3V 1 + V 4 | = 2 Det | V 2 | = 0 | V 3 | | 3V 1 + V 4 |

8. Page 266 Problem 46 Find the determinant of the linear transformation T( f ( t ) ) = f( 3 t - 2 ) | t 2 t 1 ------+---------------------------------- t 2 | 9 - 12 4 | t | 0 3 - 2 | 1 | 0 0 1 Det [T] = 27

9. Page 266 Problem 48 Find the determinant of the linear transformation L(A) = A T from R 2x2  R 2x2 | E 11 E 12 E 21 E 22 --------+-------------------------------------- E 11 | 1 0 0 0 | E 12 | 0 0 1 0 | E 21 | 0 1 0 0 | E 22 | 0 0 0 1 Determinant = - 1.

10. Page 269 Example 1. What are the possible values of the determinant of an orthogonal matrix A? Answer 1 or - 1. 1 = Det [ I ] = Det[ A A T ] = Det [ A ] Det [ A T ] = Det [ A ] 2 Therefore Det [ A ] = +1 or - 1.

11. For the plane or 3 dimensions, those with determinant +1 are rotations. Those with determinant -1 involve reflections.

12. Page 270 Example 2. I really am not sure I grasp what the book is presenting. I presume it is this. _______________ _ _/ \ / \ / \ \ / \ U/ \ / \ / \_ _ _ / _ _ __\ /__________W___/ / \ / \ / V \ / \ / _\| / \ / \--------------------/

13. | U | Det | V | is the volume of the | W | parallelepiped. It is positive if UVW is right handed. It is negative if UVW is left handed.

14. Remark. The equation of a line through (x 1,y 1 ) and (x 2,y 2 ) is | x y 1 | | x 1 y 1 1 | = 0 | x 2 y 2 1 |

15. The equation of a plane through three points (x 1,y 1,z 1 ), (x 2,y 2,z 2 ), (x 3,y 3,z 3 ) is; | x y z 1 | | x 1 y 1 z 1 1 | = 0 | x 2 y 2 z 2 1 | | x 3 y 3 z 3 1 | Notice that this is a linear function of x,y,z. It is of the form Ax+By+Cz+D = 0.

16. Notice that at points (x i,y i,z i ) the determinant is zero. Thus it must be the plane through those three points.

17. Page 274. Find the area of the parallelogram with sides (1,1,1) and (1,2,3). |1 1 1|| 1 1 | Area = Sqrt[ A T A ] = Sqrt[ Det |1 2 3|| 1 2 | ] | 1 3 | = Sqrt [ Det | 3 6 | ] = Sqrt[ 6 ] | 6 14 |

18. Page 277. The determinant represents the stretching factor of a Matrix. This explains why Det(A B) = Det(A) Det(B). This also explains why Det(A -1 ) = 1/Det[ A ].

19. Page 288. Cramer's Rule. A X = B. | a 11... a 1i... a 1n || x 1 | | b 1 | | a 21... a 2i... a 2n || x 2 | | b 2 | | a 31... a 3i... a 3n || x 3 | | b 3 | |...... || | | | | a n1... a ni... a nn || x n | | b n |

20. This means that a 11 x 1 +... + a 1i x i +... + a 1n x n = b 1 a 21 x 1 +... + a 2i x i +... + a 2n x n = b 2 a 31 x 1 +... + a 3i x i +... + a 3n x n = b 3... a n1 x 1 +... + a ni x i +... + a nn x n = b n

21. Now Replace the B column into the original matrix. | a 11... b 1... a 1n | | a 21... b 2... a 2n | | a 31... b 3... a 3n | |...... | | a n1... b n... a nn |

22. |a 11... a 11 x 1 +...+a 1i x i +...+a 1n x n... a 1n | |a 21... a 12 x 1 +...+a 2i x i +...+a 2n x n... a 2n | Det |a 31... a 31 x 1 +...+a 3i x i +...+a 3n x n... a 3n | |...... | |a n1... a n1 x 1 +...+a ni x i +...+a nn x n... a nn |  ---------------B----------- 

23. Using the rules of determinants, we have | a 11... a 1i x i... a 1n | | a 21... a 2i x i... a 2n | Det | a 31... a 3i x i... a 3n | |...... | | a n1... a ni x i... a nn |

24. x i Det[ A ]. This is Cramer’s rule. x i = Det( A i ) / Det(A) Where A i is A with column i replaced by the RHS = B.

25. Fill out the following table for the cross product. N | S | E | W | U | D ------------|-----|------|-----|------|----- N______|___|___|___|____|___ S______|___|___|___|____|___ E______|___|___|___|____|___ W______|__|___|____|____|___ U______|___|___|___|____|___ D______|___|___|___|____|___