lecture 4 - Part Three Linear Algebraic Equations Fig PT3.5...

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Part Three Linear Algebraic Equations
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Fig PT3.5
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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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Solved single equations previously Now consider more than one variable and more than one equation   0 x f 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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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 11 a a a a a a a a a a a a a a a a a a a a a a a a a A Matrix Notations Column 4 Row 3 (second index) (first index)
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Scalars, Vectors, Matrices MATLAB treat variables as “matrices” Matrix (m n) - a set of numbers arranged in rows (m) and columns (n) Scalar : 1 1 matrix Row Vector : 1 n matrix ( [b] or b ) Column Vector : m 1 matrix ( [c] or {c} )   2 2 7 1 5 0 5 9 2 3 4 2 5 2 3 1 D 21 7 3 2 02 5 b b c 21 7 3 2 02 5 b b 27 5 a T . . . . ] [ . . . ] [ } { ] [ . . . ] [ . ] [
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Square matrix, m = n Particularly important when solving simultaneous equations in engineering applications a ii principle or main diagonal 44 43 42 41 34 33 32 31 24 23 22 21 14 13 12 11 a a a a a a a a a a a a a a a a A ] [ Square Matrix
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Transpose In MATLAB, transpose is A Trace is sum of diagonal elements In MATLAB, trace(A) 44 43 42 41 34 33 32 31 24 23 22 21 14 13 12 11 a a a a a a a a a a a a a a a a A ] [ 44 34 24 14 43 33 23 13 42 32 22 12 41 31 21 11 T a a a a a a a a a a a a a a a a A ] [ Matrix Operations
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Matrix Transpose 4 2 3 1 2 3 4 2 1 3 3 2 4 y x 6 3 9 4 2 6 8 4 12 2 1 3 3 2 4 y x 2 1 3 y' ; 3 2 4 x 2 1 3 y ; 3 2 4 x ) ( ) )( ( ) )( ( * *
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4 3 0 1 6 3 1 2 5 4 0 2 3 8 1 1 5 8 2 2 6 4 1
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