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ECIV 301

ECIV 301. Programming & Graphics Numerical Methods for Engineers Lecture 12 System of Linear Equations. Objectives. Introduction to Matrix Algebra Express System of Equations in Matrix Form Introduce Methods for Solving Systems of Equations Advantages and Disadvantages of each Method.

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ECIV 301

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  1. ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 12 System of Linear Equations

  2. Objectives • Introduction to Matrix Algebra • Express System of Equations in Matrix Form • Introduce Methods for Solving Systems of Equations • Advantages and Disadvantages of each Method

  3. Matrix Algebra Rectangular Array of Elements Represented by a single symbol [A]

  4. Row 1 Row 3 Column m Column 2 Matrix Algebra n x m Matrix

  5. 3rd Row 2nd Column Matrix Algebra

  6. Matrix Algebra 1 Row, m Columns Row Vector

  7. Matrix Algebra n Rows, 1 Column Column Vector

  8. Main Diagonal Matrix Algebra If n = m Square Matrix e.g. n=m=5

  9. Matrix Algebra Special Types of Square Matrices Symmetric: aij = aji

  10. Matrix Algebra Special Types of Square Matrices Diagonal: aij = 0, ij

  11. Matrix Algebra Special Types of Square Matrices Identity: aii=1.0 aij = 0, ij

  12. Matrix Algebra Special Types of Square Matrices Upper Triangular

  13. Matrix Algebra Special Types of Square Matrices Lower Triangular

  14. Matrix Algebra Special Types of Square Matrices Banded

  15. Matrix Operating Rules - Equality [A]mxn=[B]pxq n=p m=q aij=bij

  16. Matrix Operating Rules - Addition [C]mxn= [A]mxn+[B]pxq n=p cij = aij+bij m=q

  17. Matrix Operating Rules - Addition Properties [A]+[B] = [B]+[A] [A]+([B]+[C]) = ([A]+[B])+[C]

  18. Multiplication by Scalar

  19. m=p Matrix Multiplication [A] n x m . [B] p x q = [C] n x q

  20. Matrix Multiplication

  21. Matrix Multiplication

  22. Matrix Multiplication Example

  23. Matrix Multiplication - Properties If dimensions suitable Associative: [A]([B][C]) = ([A][B])[C] Distributive: [A]([B]+[C]) = [A][B]+[A] [C] Attention: [A][B] [B][A]

  24. Operations - Transpose

  25. Operations - Inverse [A] [A]-1 [A] [A]-1=[I] If [A]-1 does not exist [A] is singular

  26. Operations - Trace Square Matrix tr[A] = Saii

  27. Linear Equations in Matrix Form

  28. Linear Equations in Matrix Form

  29. Linear Equations in Matrix Form

  30. Linear Equations in Matrix Form

  31. Homework Problems 9.1, 9.2, 9.3 Due Date: Oct 6

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