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This table summarizes the factors affecting the size of a general solution. number of solutions of the homogeneous system one infinitely many infinitely many particular yes solution unique solution solutions exists? no no no solutions solutions The dimension on the top of the table is the simpler one. When we perform Gauss's Method on a linear system, ignoring the constants on the right side and so paying attention only to the coefficients on the left-hand side, we either end with every variable leading some row or else we find some variable that does not lead a row, that is, we find some variable that is free. (We formalize "ignoring the constants on the right" by considering the associated homogeneous system.) A notable special case is systems having the same number of equations as unknowns. Such a system will have a solution, and that solution will be unique, if and only if it reduces to an echelon form system where every variable leads its row (since there are the same number of variables as rows), which will happen if and only if the associated homogeneous system has a unique solution. 3.11 Definition A square matrix is nonsingular if it is the matrix of coefficients of a homogeneous system with a unique solution. It is singular otherwise, that is, if it is the matrix of coefficients of a homogeneous system with infinitely many solutions.

This table summarizes the factors affecting the size of a general solution. number of solutions of the homogeneous system one infinitely many infinitely many particular yes solution unique solution solutions exists? no no no solutions solutions The dimension on the top of the table is the simpler one. When we perform Gauss's Method on a linear system, ignoring the constants on the right side and so paying attention only to the coefficients on the left-hand side, we either end with every variable leading some row or else we find some variable that does not lead a row, that is, we find some variable that is free. (We formalize "ignoring the constants on the right" by considering the associated homogeneous system.) A notable special case is systems having the same number of equations as unknowns. Such a system will have a solution, and that solution will be unique, if and only if it reduces to an echelon form system where every variable leads its row (since there are the same number of variables as rows), which will happen if and only if the associated homogeneous system has a unique solution. 3.11 Definition A square matrix is nonsingular if it is the matrix of coefficients of a homogeneous system with a unique solution. It is singular otherwise, that is, if it is the matrix of coefficients of a homogeneous system with infinitely many solutions.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Q1 1) In which of the four cases is the system consistent and in which of the four cases is the system inconsistent? (Look at the table in the middle of the page)

PROOP We've seen examples of all three happening so we need only prove that there are no other possibilities. First observe a homogeneous system with at least one non-o solution v has infinitely many solutions. This is because any scalar multiple of V also solves the homogeneous system and there are infinitely many vectors in the set of scalar multiples of V: if s, t R are unequal then sv tv, since sv-tv = (st)v is non-Ō as any non-0 component of V, when rescaled by the non-0 factor s-t, will give a non-0 value. Now apply Lemma 3.7 to conclude that a solution set {p+hh solves the associated homogeneous system) is either empty (if there is no particular solution p), or has one element (if there is a p and the homogeneous system has the unique solution 0), or is infinite (if there is a p and the homogeneous system has a non-o solution, and thus by the prior paragraph has infinitely many solutions). QED This table summarizes the factors affecting the size of a general solution. number of solutions of the homogeneous system one infinitely many infinitely many particular yes solution unique solution solutions no exists? no no solutions solutions The dimension on the top of the table is the simpler one. When we perform Gauss's Method on a linear system, ignoring the constants on the right side and so paying attention only to the coefficients on the left-hand side, we either end with every variable leading some row or else we find some variable that does not lead a row, that is, we find some variable that is free. (We formalize "ignoring the constants on the right" by considering the associated homogeneous system.) A notable special case is systems having the same number of equations as unknowns. Such a system will have a solution, and that solution will be unique, if and only if it reduces to an echelon form system where every variable leads its row (since there are the same number of variables as rows), which will happen if and only if the associated homogeneous system has a unique solution. 3.11 Definition A square matrix is nonsingular if it is the matrix of coefficients of a homogeneous system with a unique solution. It is singular otherwise, that is, if it is the matrix of coefficients of a homogeneous system with infinitely many solutions.
Transcribed Image Text:PROOP We've seen examples of all three happening so we need only prove that there are no other possibilities. First observe a homogeneous system with at least one non-o solution v has infinitely many solutions. This is because any scalar multiple of V also solves the homogeneous system and there are infinitely many vectors in the set of scalar multiples of V: if s, t R are unequal then sv tv, since sv-tv = (st)v is non-Ō as any non-0 component of V, when rescaled by the non-0 factor s-t, will give a non-0 value. Now apply Lemma 3.7 to conclude that a solution set {p+hh solves the associated homogeneous system) is either empty (if there is no particular solution p), or has one element (if there is a p and the homogeneous system has the unique solution 0), or is infinite (if there is a p and the homogeneous system has a non-o solution, and thus by the prior paragraph has infinitely many solutions). QED This table summarizes the factors affecting the size of a general solution. number of solutions of the homogeneous system one infinitely many infinitely many particular yes solution unique solution solutions no exists? no no solutions solutions The dimension on the top of the table is the simpler one. When we perform Gauss's Method on a linear system, ignoring the constants on the right side and so paying attention only to the coefficients on the left-hand side, we either end with every variable leading some row or else we find some variable that does not lead a row, that is, we find some variable that is free. (We formalize "ignoring the constants on the right" by considering the associated homogeneous system.) A notable special case is systems having the same number of equations as unknowns. Such a system will have a solution, and that solution will be unique, if and only if it reduces to an echelon form system where every variable leads its row (since there are the same number of variables as rows), which will happen if and only if the associated homogeneous system has a unique solution. 3.11 Definition A square matrix is nonsingular if it is the matrix of coefficients of a homogeneous system with a unique solution. It is singular otherwise, that is, if it is the matrix of coefficients of a homogeneous system with infinitely many solutions.
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