2 Theorem 2 3 If A is a homogeneous system of m linear equations with n

# 2 theorem 2 3 if a is a homogeneous system of m

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2 Theorem 2 . 3 If [ A | 0 ] is a homogeneous system of m linear equations with n variables, where m < n , then the system has infinitely many solutions. Proof. After applying Gauss-Jordan Elimination to [ A | 0 ] , we get [ B | 0 ] in reduced row echelon form. We make two observations on [ B | 0 ] : (1) [ B | 0 ] is consistent: If B has an all-zeros row, then that row corresponds to the equation 0 = 0 because it is homogenous. So, it has at least one solution. (2) It has at least one free variable: Since B has m equations, it can at most have m non-zero rows. Hence, it has at least n - m > 0 free variables.
§ 1.1 and § 1.2 1.26 Homogeneous Systems 1 Definition A system of linear equations is called homogeneous of the constant term in each equation is zero. Otherwise, it is called non - homogeneous . If the [ A | b ] is homogeneous, then b = 0 . 2 Theorem 2 . 3 If [ A | 0 ] is a homogeneous system of m linear equations with n variables, where m < n , then the system has infinitely many solutions. Proof. After applying Gauss-Jordan Elimination to [ A | 0 ] , we get [ B | 0 ] in reduced row echelon form. We make two observations on [ B | 0 ] : (1) [ B | 0 ] is consistent: If B has an all-zeros row, then that row corresponds to the equation 0 = 0 because it is homogenous. So, it has at least one solution. (2) It has at least one free variable: Since B has m equations, it can at most have m non-zero rows. Hence, it has at least n - m > 0 free variables. ( 1 ) and ( 2 ) together imply that the system has infinitely many solutions. (each free variable introduces a new variable in the solution set)
§ 1.1 and § 1.2 1.27 What if it is non-homogeneous? The last row in the following SLEs gives 0 = 1. This is not possible. So, it has no solution. 1 0 1 0 1 2 0 0 1
§ 1.1 and § 1.2 1.27 What if it is non-homogeneous? The last row in the following SLEs gives 0 = 1. This is not possible. So, it has no solution. 1 0 1 0 1 2 0 0 1 Remark 1 : An SLEs has NO solution (inconsistent), if its augmented matrix [ A | b ] is row equivalent to some [ B | c ] such that one of the rows of [ B | c ] has the form £ 0 0 ... 0 0 d 6= 0 /
§ 1.1 and § 1.2 1.27 What if it is non-homogeneous? The last row in the following SLEs gives 0 = 1. This is not possible. So, it has no solution. 1 0 1 0 1 2 0 0 1 Remark 1 : An SLEs has NO solution (inconsistent), if its augmented matrix [ A | b ] is row equivalent to some [ B | c ] such that one of the rows of [ B | c ] has the form £ 0 0 ... 0 0 d 6= 0 / Remark 2 : It is enough to apply elementary row operations and Gauss-Jordan Elimination to [ A | b ] . The process will itself tell you whether the system is consistent or inconsistent.
§ 1.1 and § 1.2 1.28 CAUTION THE QUIZ WILL COVER EVERYTHING IN 1.1 AND 1.2 EXCLUDING REDUCED ROW ECHELON FORMS AS WE HAVE NOT COVERED IT YET.