Lecture13 - Engineering Analysis ENG 3420 Fall 2009 Dan C...

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Unformatted text preview: Engineering Analysis ENG 3420 Fall 2009 Dan C. Marinescu Office: HEC 439 B Office hours: Tu-Th 11:00-12:00 2 Lecture 13 Lecture 13 Last time: Problem solving in preparation for the quiz Linear Algebra Concepts Vector Spaces, Linear Independence Orthogonal Vectors, Bases Matrices Today Solving systems of linear equations (Chapter 9) Graphical methods Next Time Gauss elimination Solving systems of linear equations Matrices provide a concise notation for representing and solving simultaneous linear equations: a 11 x 1 + a 12 x 2 + a 13 x 3 = b 1 a 21 x 1 + a 22 x 2 + a 23 x 3 = b 2 a 31 x 1 + a 32 x 2 + a 33 x 3 = b 3 a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 x 1 x 2 x 3 = b 1 b 2 b 3 [ A ]{ x } = { b } Solving systems of linear equations in Matlab Two ways to solve systems of linear algebraic equations [A]{x}={b}: Left-division x = A\b Matrix inversion x = inv(A)*b Matrix inversion only works for square, non-singular systems; it is less efficient than left-division. Solving graphically systems of linear equations For small sets of simultaneous equations, graphing them and determining the location of the intersection of the straight line representing each equation provides a solution . There is no guarantee that one can find the solution of system of linear equations: a) No solution exists b) Infinite solutions exist c) System is ill-conditioned Determinant of the square matrix A=[a ij ] Here the coefficient A ij of a ij is called the cofactor of A A cofactor is a polynomial in the remaining rows of A and can be described as the partial derivative of A. The cofactor polynomial described as the partial derivative of A....
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Lecture13 - Engineering Analysis ENG 3420 Fall 2009 Dan C...

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