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Math 307 Spring, 2003 Hentzel Time: 1:10-2:00 MWF Room: 1324 Howe Hall

Explore applications and properties of determinants in linear algebra, including Cramer’s Rule and matrix transformations.

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Math 307 Spring, 2003 Hentzel Time: 1:10-2:00 MWF Room: 1324 Howe Hall

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  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. | 3V1 + V4 | = 2 Det | V2 | = 0 | V3 | | 3V1 + V4 |

  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 | E11 E12 E21 E22 --------+-------------------------------------- E11 | 1 0 0 0 | E12 | 0 0 1 0 | E21 | 0 1 0 0 | E22 | 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 (x1,y1) and (x2,y2) is | x y 1 | | x1 y1 1 | = 0 | x2 y2 1 |

  15. The equation of a plane through three points (x1,y1,z1), (x2,y2,z2), (x3,y3,z3) is; | x y z 1 | | x1 y1 z1 1 | = 0 | x2 y2 z2 1 | | x3 y3 z3 1 | Notice that this is a linear function of x,y,z. It is of the form Ax+By+Cz+D = 0.

  16. It is of the form Ax+By+Cz+D = 0. Notice that at points (xi,yi,zi) 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. | a11 ... a1i ... a 1n || x1 | | b1 | | a21 ... a2i ... a 2n || x2 | | b2 | | a31 ... a3i ... a 3n || x3 | | b3 | | ... ... || | | | | ... ... || | | | | an1 ... ani ... a nn || xn | | bn |

  20. This means that a11 x1 + ... + a1i xi + ... + a1n xn = b1 a21 x1 + ... + a2i xi + ... + a2n xn = b2 a31 x1 + ... + a3i xi + ... + a3n xn = b3 . . . . . . an1 x1 + ... + ani xi + ... + ann xn = bn

  21. Now Replace the B column into the original matrix. | a11 ... b1 ... a 1n | | a21 ... b2 ... a 2n | | a31 ... b3 ... a 3n | | ... ... | | ... ... | | an1 ... bn ... a nn |

  22. |a11 ... a11x1+...+a1ixi+...+a1nxn ... a1n| |a21 ... a12x1+...+a2ixi+...+a2nxn ... a2n| Det |a31 ... a31x1+...+a3ixi+...+a3nxn ... a3n| | ... ... | | ... ... | |an1 ... an1x1+...+anixi+...+annxn ... ann| ---------------B-----------

  23. Using the rules of determinants, we have | a11 ... a1ixi ... a1n | | a21 ... a2ixi ... a2n | Det | a31 ... a3ixi ... a3n | | ... ... | | ... ... | | an1 ... anixi ... ann |

  24. xi Det[ A ]. This is Cramer’s rule. xi = Det( Ai ) / Det(A) Where Ai 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______|___|___|___|____|___

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