Notes 3 GenSolLinSys 2009

Notes 3 GenSolLinSys 2009 - File: Notes 3 GenSolLinSys...

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1 File: Notes 3 GenSolLinSys 2009.Doc General Solutions to Linear Systems of Equations Consider the problem of obtaining a solution to the system of equations a x a x a x b ax ax ax b nn mm m n n m 11 1 12 2 1 1 21 1 22 2 2 2 11 2 2 + + + = ++ += " " # " which we represent in matrix form as Ax b = , where A is an mn × matrix, b is a vector of dimension m , and x is a vector of n unknowns. Assuming that the rank of A is m < n , and then there are an infinity of solutions to this system of equations. The rank of a matrix A is the largest nonzero determinant which can be created from the matrix A by omitting some rows and columns of A . More generally, the system has a solution if the rank of A and the rank of the A matrix augmented with b , denoted A b , have the same rank. (Note that we will use the name rank ( ) to designate this matrix function.) That is, () ( ) rank A rank Ab = or 11 12 1 11 12 1 1 21 22 21 2 2 12 m m mn m m mn m aa a ab a a a rank rank a ⎡⎤ ⎢⎥ = ⎣⎦ "" ## . If the rank ( A ) = rank ( A b ) = m , and m < n , then there are an infinity of solutions to the system. If the rank ( A ) = rank ( A b ) = m and m = n , then there is only one solution to the system of equations. Lastly, if the rank ( A ) < rank ( A b ), then there is no solution to the system of equations. From here on, we will assume that there are an infinite number of solutions to the system of equations. To obtain a general solution to the system of equations, we partition the system into two sets of variables and their respective coefficients. To motivate this partition, consider for a moment that the matrix A is square, and thus of rank m . Then the, unique, solution would be 1 x Ab = . Note that being of rank m implies that the determinant of A is nonzero and, hence, the inverse exists. Now consider that the system consists of exactly m + 1 variables, with the coefficients of the first
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This note was uploaded on 10/26/2011 for the course ISEN 620 taught by Professor Curry during the Fall '10 term at Texas A&M.

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Notes 3 GenSolLinSys 2009 - File: Notes 3 GenSolLinSys...

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