201-Lect 3 - Outline Week 3 What are linear algebraic equations Matrix algebra overview Solving linear algebraic equations graphics methods direct

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1 Week 3 Week 3 Linear Algebraic Equations Week 3 Outline • What are linear algebraic equations • Matrix algebra overview • Solving linear algebraic equations - graphics methods - direct methods: Gauss elimination LU decomposition Thomas methods Week 3 Linear Algebraic Equations • Background - Single-component systems result in a single equation that can be solved using root-location techniques as discussed in week 2 - Multicomponent systems result in a coupled set of mathematical equations that must be solved simultaneously. The equations are coupled because the individual parts of the system are influenced by other parts Week 3 Engineering Systems • Mechanical (Spring-mass system) = + + + F x x x x k k k k k k k k k k k k k 0 0 0 0 0 0 0 0 0 4 3 2 1 4 4 4 4 3 3 3 3 2 2 2 2 1
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2 Week 3 Engineering Systems • Electrical (circuit) Week 3 Engineering Systems • Civil/Environmental Week 3 Linear Algebraic Equations • General form m m nm n n m m m m b x a x a x a b x a x a x a b x a x a x a = + + + = + + + = + + + L M M M M M L L 2 2 1 1 2 2 2 22 1 21 1 1 2 12 1 11 the a ’s are constant coefficients, the b ’s are constants, the x ’s are unknows, and n is the number of equations n , , , i b x a m j i j ij L 2 1 1 = = = Week 3 Linear Algebraic Equations • Matrix notation matrix A , vector x and vector b n = m , square matrix A = m m nm n n m m b b b x x x a a a a a a a a a M M L M O M M L L 2 1 2 1 2 1 2 22 21 1 12 11 A n × m x b
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3 Week 3 Linear Algebraic Equations • Simple example: 2 × 2 System of Linear equations (SL) 4 3 10 2 4 2 1 2 1 = + = + x x x x = 4 10 3 1 2 4 2 1 x x notation matrix The solution to the above SL is x 1 = 1.57142854 and x 2 = 1.85714281 or the vector x , = 85714281 1 57142854 1 . . x Week 3 Matrix Algebra • Square matrix = nn n n n a a a an a a a a a A L M O M M L L 2 1 22 21 1 12 11 • Diagonal matrix = = = j i for a j i for a ij ij nxn nn a a a D 0 0 0 0 0 0 0 0 22 11 L M O M M L L 4 x 4 5 . 0 0 0 0 0 1 0 0 0 0 1 . 3 0 0 0 0 5 . 2 Week 3 An important property of identity matrix is that any matrix multiplied by an identity matrix will yield the same matrix. I A = A Matrix Algebra • Identity matrix • Triangular matrix 4 x 4 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 4 4 4 0 0 0 4 2 1 0 0 0 7 2 5 1 0 1 5 4 2 3 x U = . .
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This note was uploaded on 10/27/2009 for the course MECH 201 taught by Professor Weihuali during the Three '09 term at University of Wollongong, Australia.

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201-Lect 3 - Outline Week 3 What are linear algebraic equations Matrix algebra overview Solving linear algebraic equations graphics methods direct

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