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ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 14 Elimination Methods

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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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Last Time Matrix Algebra Rectangular Array of Elements Represented by a single symbol [A]

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Last Time Matrix Algebra Row 1 Row 3 Column 2Column m n x m Matrix

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Last Time Matrix Algebra 3 rd Row 2 nd Column

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Last Time Matrix Algebra 1 Row, m Columns Row Vector

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Last Time Matrix Algebra n Rows, 1 Column Column Vector

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

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Last Time Matrix Algebra Special Types of Square Matrices Symmetric: a ij = a ji

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Last Time Matrix Algebra Diagonal: a ij = 0, i j Special Types of Square Matrices

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Last Time Matrix Algebra Identity: a ii =1.0 a ij = 0, i j Special Types of Square Matrices

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Last Time Matrix Algebra Upper Triangular Special Types of Square Matrices

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Last Time Matrix Algebra Lower Triangular Special Types of Square Matrices

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Last Time Matrix Algebra Banded Special Types of Square Matrices

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Last Time Matrix Operating Rules - Equality [A] mxn =[B] pxq n=pm=qa ij =b ij

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Last Time Matrix Operating Rules - Addition [C] mxn = [A] mxn +[B] pxq n=p m=q c ij = a ij +b ij

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Last Time Multiplication by Scalar

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

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Last Time Matrix Multiplication

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Last Time Operations - Transpose

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Last Time Operations - Inverse [A][A] -1 [A] [A] -1 =[I] If [A] -1 does not exist [A] is singular

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Last Time Operations - Trace Square Matrix tr[A] = a ii

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Equations in Matrix Form Consider

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Linear Equations in Matrix Form

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# Equations = # Unknowns = n Square Matrix n x n

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Solution of Linear Equations Consider the system

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Solution of Linear Equations

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Express In Matrix Form Upper Triangular What is the characteristic? Solution by Back Substitution

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Solution of Linear Equations Objective Can we express any system of equations in a form 0

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Background Consider (Eq 1) (Eq 2) Solution 2*(Eq 1) (Eq 2) Solution !!!!!! Scaling Does Not Change the Solution

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Background Consider (Eq 1) (Eq 2)-(Eq 1) Solution !!!!!! (Eq 1) (Eq 2) Solution Operations Do Not Change the Solution

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Gauss Elimination Example Forward Elimination

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Gauss Elimination -

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Substitute 2 nd eq with new

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Gauss Elimination -

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Substitute 3rd eq with new

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Gauss Elimination -

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Substitute 3rd eq with new

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Gauss Elimination

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