Lecture 5 - Chapter 8 Linear Algebraic Equations and...

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Chapter 8 Linear Algebraic Equations and Matrices
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Three individuals connected by bungee cords
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Free-body diagrams Newton’s second law = - - = - - - + = - - + 0 x x k g m 0 x x k x x k g m 0 x k x x k g m 2 3 3 3 1 2 2 2 3 3 2 1 1 1 2 2 1 ) ( ) ( ) ( ) ( = + - = - + - = - + g m x k x k g m x k x k k x k g m x k x k k 3 3 3 2 3 2 3 3 2 3 2 1 2 1 2 2 1 2 1 ) ( ) ( Rearrange the equations [ K ] { x } = { b }
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Newton’s second law – equation of motion Kirchhoff’s current and voltage rules Mass-spring system (similar to bungee jumpers) Resistor circuits
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Solved single equations previously Now consider more than one variable and more than one equation ( 29 0 x f = ( 29 ( 29 ( 29 0 x ,..., x , x f 0 x ,..., x , x f 0 x ,..., x , x f n 2 1 n n 2 1 2 n 2 1 1 = = = Linear Algebraic Equations
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Linear equations and constant coefficients a ij and b i are constants n n nn 2 2 n 1 1 n 2 n n 2 2 22 1 21 1 n n 1 2 12 1 11 b x a ... x a x a b x a ... x a x a b x a ... x a x a = + + + = + + + = + + + Linear Systems
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} { } ]{ [ ] [ ] ][ [ b x A or b x A = = = n 2 1 n 2 1 nn 2 n 1 n n 2 22 21 n 1 12 11 b b b x x x a a a a a a a a a Mathematical background It is convenient to write system of equations in matrix-vector form
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[ ] = mn 4 m 3 m 2 m 1 m n 4 44 43 42 41 n 3 34 33 32 31 n 2 24 23 22 21 n 1 14 13 12
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This note was uploaded on 11/15/2010 for the course EGM 5403 taught by Professor Mei during the Spring '10 term at University of Florida.

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Lecture 5 - Chapter 8 Linear Algebraic Equations and...

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