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Lecture 4 Existence and Uniqueness of Solutions The Inverse Conditions for Invertibility AMath 352 Mon Apr 5 1 16 x 2y 3 2x 4y 1 x 2y 3 2x 4y 6 The left set of equations has no solutions and the right set of equations has in nitely many solutions Set y to any value and take x 3 2y Existence and Uniqueness of Solutions Under what conditions does a system of m linear equations in n unknowns have a solution and under what conditions is the solution unique 2 16 Existence and Uniqueness of Solutions Under what conditions does a system of m linear equations in n unknowns have a solution and under what conditions is the solution unique x 2y 3 2x 4y 1 x 2y 3 2x 4y 6 The left set of equations has no solutions and the right set of equations has in nitely many solutions Set y to any value and take x 3 2y 2 16 1 4 7 2 5 8 3 6 9 0 1 c 1 0 2 0 0 3 6 3 0 1 0 c 2 1 2 3 0 3 6 0 6 12 0 1 c No solution if c cid 54 2 in nitely many solutions if c 2 Set z to anything and solve for x and y Existence and Uniqueness of Solutions Cont It is less obvious when there are more equations and unknowns x 2y 3z 0 4x 5y 6z 1 7x 8y 9z c 3 16 Existence and Uniqueness of Solutions Cont It is less obvious when there are more equations and unknowns x 2y 3z 0 4x 5y 6z 1 7x 8y 9z c 1 2 4 5 7 8 3 6 9 0 1 c 3 1 2 0 3 6 0 6 12 0 1 c 1 2 0 3 6 0 0 3 0 0 1 c 2 No solution if c cid 54 2 in nitely many solutions if c 2 Set z to anything and solve for x and y 3 16 Now consider a linear system A x b where b cid 54 cid 126 0 If this has one solution x and if the homogeneous system A cid 126 x cid 126 0 has a nonzero solution cid 126 y then this linear system has in nitely many solutions since A x cid 126 y A x A cid 126 y b cid 126 0 b Homogeneous Linear Systems Consider a homogeneous linear system A cid 126 x cid 126 0 Certainly cid 126 x cid 126 0 is one solution Suppose there is another solution cid 126 y whose entries are not all 0 Then there are in nitely many solutions since for any scalar A cid 126 y A cid 126 y cid 126 0 cid 126 0 4 16 Homogeneous Linear Systems Consider a homogeneous linear system A cid 126 x cid 126 0 Certainly cid 126 x cid 126 0 is one solution Suppose there is another solution cid 126 y whose entries are not all 0 Then there are in nitely many solutions since for any scalar A cid 126 y A cid 126 y cid 126 0 cid 126 0 Now consider a linear system A x b where b cid 54 cid 126 0 If this has one solution x and if the homogeneous system A cid 126 x cid 126 0 has a nonzero solution cid 126 y then this linear system has in nitely many solutions since A x cid 126 y A x A cid 126 y b cid 126 0 b 4 16 Theorem Let A be an m by n matrix The homogeneous system A cid 126 x cid 126 0 either has only the trivial solution cid 126 x cid 126 0 or it has in nitely many solns If b cid 54 cid 126 0 is an m vector then the inhomogeneous system A x b has 1 No solns if b is not in the range of A 2 Exactly one soln if b is in the range of A and the homogeneous system has only the trivial soln cid 126 x cid 126 0 3 In nitely many solns if b is in the range of A and the homogeneous system has a nontrivial soln cid 126 y cid 54 cid 126 0 Range of a Matrix No of Solutions of a Linear System De nition The range of A is the set of all vectors that can be written in the form A cid 126 x for some vector cid 126 x 5 16 Range of a Matrix No of Solutions of a Linear System De nition The range of A is the set of all vectors that can be written in the form A cid 126 x for some vector cid 126 x Theorem Let A be an m by n matrix The homogeneous system A cid 126 x cid 126 0 either has only the trivial solution cid 126 x cid 126 0 or it has in nitely many solns If b cid 54 cid 126 0 is an m vector then the inhomogeneous system A x b has 1 No solns if b is not in the range of A 2 Exactly one soln if b is in the range of A and the homogeneous system has only the trivial soln cid 126 x cid 126 0 3 In nitely many solns if b is in the range of A and the homogeneous system has a nontrivial soln cid 126 y cid 54 cid 126 0 5 16 Let cid 126 a1 cid 126 a2 cid 126 an denote the columns of A Claim A cid 126 x x1 cid 126 a1 x2 cid 126 a2 xn cid 126 an since this is just cid 80 n cid 80 n j 1 a1j xj j 1 a2j xj cid 80 n j 1 amj xj Another View of Matrix Vector Multiplication a12 a11 a22 a21 am1 am2 a1n a2n amn x1 x2 xn cid 80 n cid 80 n j 1 a1j xj j 1 a2j xj j 1 amj xj cid 80 n 6 16 Another View of Matrix Vector Multiplication a12 a11 a22 a21 am1 am2 a1n a2n amn x1 x2 xn cid 80 n cid 80 n j 1 a1j xj j 1 a2j xj j 1 amj xj cid 80 n Let cid 126 a1 cid 126 a2 cid 126 an denote the columns of A Claim A cid 126 x x1 cid 126 a1 x2 cid 126 a2 xn cid 126 an since this is just cid 80 n cid 80 n j 1 a1j xj j 1 a2j xj j 1 amj xj cid 80 n 6 16 Another View of Matrix Vector Multiplication Cont Example 1 2 3 4 5 6 7 8 9 1 2 1 1 1 2 2 3 1 4 1 5 2 6 1 7 1 8 2 9 1 6 12 18 1 2 1 3 6 9 6 12 18 1 4 7 2 5 8 7 16 Since …

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