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Unformatted text preview: E7 L14 1 E7: INTRODUCTION TO COMPUTER E7: INTRODUCTION TO COMPUTER PROGRAMMING FOR SCIENTISTS AND PROGRAMMING FOR SCIENTISTS AND ENGINEERS ENGINEERS Lecture Outline (pages 358376) 1. Linear algebraic equations 2. Basic matrix algebra 3. Solving linear algebraic equations with MATLAB (# of unknowns = # of equations) Copyright 2007, Horowitz, Packard. This work is licensed under the Creative Commons AttributionShare Alike License. To view a copy of this license, visit http://creativecommons.org/licenses/bysa/2.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA. E7 L14 2 Linear algebraic equations Linear algebraic equations Many engineering and science problems reduce to the simultaneous solution of several linear algebraic equations. Some typical examples: • Curvefitting of data • Governing physical phenomena such as • deformation of solids, flow of liquids, conduction of heat and electricity, etc. • Design of engineering systems – Example: simple truss structures • Consist of beams • Frictionless “pin” joints E7 L14 3 Linear algebraic equations Linear algebraic equations Consider n LINEAR equations and m unknowns. m unknowns n equations 11 12 1 1 21 22 2 2 1 1 2 2 2 1 1 2 m m n n m m n m m n A A A b A A A b A A A b x x x x x x x x x + + + = + + + = + + + = L L M L E7 L14 4 Linear algebraic equations Linear algebraic equations Consider n LINEAR equations and m unknowns. Think of the A ij ’s and b i ’s as known numbers j 11 12 1 1 21 22 2 2 1 1 2 2 2 1 1 2 m m n n m m n m m n A A A b A A A b A A A b x x x x x x x x x + + + = + + + = + + + = L L M L E7 L14 5 Example: intersection of two lines on a plane Example: intersection of two lines on a plane 1 2 3 4 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 intersection E7 L14 6 Example: intersection of two lines on a plane Example: intersection of two lines on a plane • The intersection of the two lines is the solution of: 2 equations 2 unknowns: x 1 and x 2 Gauss elimination is an efficient method for solving linear algebraic equations when the number of equations n is equal to the number of unknowns m E7 L14 7 Gauss elimination method Gauss elimination method n = 2 equations m = 2 unknowns • Multiply one equation by a factor and add to the other equation to eliminate one variable E7 L14 8 Solution using Gauss elimination method Solution using Gauss elimination method E7 L14 9 Solution using Gauss elimination method Solution using Gauss elimination method pivot add first equation to second equation result E7 L14...
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 Spring '08
 Patzek
 Linear Algebra, Invertible matrix, Packard, Det

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