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Lecture 6 lu factorization & determinants - section 2-5 2-7 3-1 and 3-2

MATH 337-102 Lecture 2

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Lecture 6 lu factorization & determinants - section 2-5 2-7 3-1 and 3-2

  1. 1. © 2012 Pearson Education, Inc. Math 337-102 Lecture 6 LU Factorization Computer Graphics Determinants
  2. 2. Slide 2.2- 2© 2012 Pearson Education, Inc. LU Factorization  Factor A = LU  A is mxn  A does NOT have to be square!!!  L is mxm (square) – Lower Triangular with 1’s on the diagonal  U is mxn  REF(A)
  3. 3. Using LU to Solve Ax = b  Ax = b  (LU)x = b  L(Ux) = b  Let y = Ux  Solve Ly = b  Then Ux = y Slide 2.2- 3© 2012 Pearson Education, Inc.
  4. 4. LU Example Slide 2.2- 4© 2012 Pearson Education, Inc.
  5. 5. Finding L and U  U is Row Echelon Form of A using row replacement only  No interchanges or scaling!!!!  L entries are such that the same sequence of row operations that reduce A to U will reduce L to I. Slide 2.2- 5© 2012 Pearson Education, Inc.
  6. 6. Finding L and U - Example Slide 2.2- 6© 2012 Pearson Education, Inc.
  7. 7. Using LU to solve Ax = b  Factor A as LU  Solve Ly = b for y  Solve Ux = y for x Slide 2.2- 7© 2012 Pearson Education, Inc.
  8. 8. Computer Graphics
  9. 9. Computer Graphics Slide 2.2- 9© 2012 Pearson Education, Inc.
  10. 10. Translations – Homogeneous Coordinates Slide 2.2- 10© 2012 Pearson Education, Inc.
  11. 11. Determinants Slide 2.2- 11© 2012 Pearson Education, Inc.
  12. 12. Determinants – Co-Factors Slide 2.2- 12© 2012 Pearson Education, Inc.
  13. 13. Determinants by Co-Factor Expansion Slide 2.2- 13© 2012 Pearson Education, Inc.
  14. 14. Cofactor Expansion Example Slide 2.2- 14© 2012 Pearson Education, Inc.
  15. 15. Cofactor Expansion Example Slide 2.2- 15© 2012 Pearson Education, Inc.
  16. 16. Determinants of Triangular Matrices Slide 2.2- 16© 2012 Pearson Education, Inc.
  17. 17. Row Operations and Determinants  Thm 3-3: Let A be a square matrix. a)If a multiple of one row is added to another row to produce a matrix B, then |B| = |A| b)If two rows are interchanged to produce B, then |B| = -|A| c)If one row of A is multiplied by k to produce B, then |B| = k|A| Slide 2.2- 17© 2012 Pearson Education, Inc.
  18. 18. Row Operations and Determinants  Row replacement does not change determinant  Row interchange negates the determinant  Scaling – think of as factoring Slide 2.2- 18© 2012 Pearson Education, Inc.

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