System - ME 210 Applied Mathematics for Mechanical...

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ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010 System of Linear Algebraic Equations 11 1 12 2 1n n 1 21 1 22 2 2n n 2 m1 1 m2 2 mn n m a x + a x + . .. + a x = b ............ a x + a x + . .. + a x = b Consider a system of m linear algebraic equations with n unknowns: x 1 , x 2 , . .. , x n where a ij and b i are constants. This system of equations can be written as a single vector-matrix equation as: [A] [x] = [b] Coefficient matrix Column vector of unknowns Column vector of known values
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ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010   mn 2 m 1 m n 2 22 21 n 1 12 11 a a a a a a a a a A   n 2 1 x x x x   m 2 1 b b b b If [b] = [0], the resulting system is called a homogeneous system of linear equations. If [b] [0], the resulting system is called a non-homogeneous system of linear equations. Coefficient matrix Column vector of unknowns Column vector of known values
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ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010 Theorem: (Existence/Uniqueness of Solution for Linear Systems) Given a linear system of m equations for n unknowns as [A] [x] = [b], where [A] is the m×n coefficient matrix, [x] is the n–dimensional unknown column vector, and [b] is the m–dimensional column vector. Let the rank of the augmented matrix [[A]¦[b]] be r. Then, such a system is called as a) inconsistent (has no solution) if rank[A] < r = rank [[A]¦[b]] , b) consistent (has at least one solution) if rank[A] = r = rank [[A]¦[b]] .
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This note was uploaded on 07/31/2011 for the course ME 210 taught by Professor Farukarinç during the Spring '09 term at Middle East Technical University.

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System - ME 210 Applied Mathematics for Mechanical...

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