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Lecture 4: Existence and Uniqueness ofSolutions, The Inverse, Conditions for InvertibilityAMath 352Mon., Apr. 51 / 16Existence and Uniqueness of SolutionsUnder what conditions does a system of m linear equations in nunknowns have a solution, and under what conditions is thesolution unique?x + 2y = 3 x + 2y = 32x + 4y = 1 2x + 4y = 6The left set of equations has no solutions and the right set ofequations has infinitely many solutions: Set y to any value andtake x = 3 − 2y.2 / 16Existence and Uniqueness of SolutionsUnder what conditions does a system of m linear equations in nunknowns have a solution, and under what conditions is thesolution unique?x + 2y = 3 x + 2y = 32x + 4y = 1 2x + 4y = 6The left set of equations has no solutions and the right set ofequations has infinitely many solutions: Set y to any value andtake x = 3 − 2y.2 / 16Existence and Uniqueness of Solutions, Cont.It is less obvious when there are more equations and unknowns:x + 2y + 3z = 04x + 5y + 6z = 17x + 8y + 9z = c,1 2 3 | 04 5 6 | 17 8 9 | c→1 2 3 | 00 −3 −6 | 10 −6 −12 | c→1 2 3 | 00 −3 −6 | 10 0 0 | c − 2.No solution if c 6= 2, infinitely many solutions if c = 2: Set z toanything and solve for x and y.3 / 16Existence and Uniqueness of Solutions, Cont.It is less obvious when there are more equations and unknowns:x + 2y + 3z = 04x + 5y + 6z = 17x + 8y + 9z = c,1 2 3 | 04 5 6 | 17 8 9 | c→1 2 3 | 00 −3 −6 | 10 −6 −12 | c→1 2 3 | 00 −3 −6 | 10 0 0 | c − 2.No solution if c 6= 2, infinitely many solutions if c = 2: Set z toanything and solve for x and y.3 / 16Homogeneous Linear SystemsConsider a homogeneous linear system A~x =~0. Certainly,~x =~0 isone solution. Suppose there is another solution~y whose entries arenot all 0. Then there are infinitely many solutions since for anyscalar α,A(α~y) = αA~y = α~0 =~0.Now consider a linear system Aˆx =ˆb, whereˆb 6=~0. If this has onesolution ˆx and if the homogeneous system A~x =~0 has a nonzerosolution~y, then this linear system has infinitely many solutionssinceA(ˆx + α~y) = Aˆx + αA~y =ˆb + α~0 =ˆb.4 / 16Homogeneous Linear SystemsConsider a homogeneous linear system A~x =~0. Certainly,~x =~0 isone solution. Suppose there is another solution~y whose entries arenot all 0. Then there are infinitely many solutions since for anyscalar α,A(α~y) = αA~y = α~0 =~0.Now consider a linear system Aˆx =ˆb, whereˆb 6=~0. If this has onesolution ˆx and if the homogeneous system A~x =~0 has a nonzerosolution~y, then this linear system has infinitely many solutionssinceA(ˆx + α~y) = Aˆx + αA~y =ˆb + α~0 =ˆb.4 / 16Range of a Matrix, No. of Solutions of a Linear SystemDefinition. The range of A is the set of all vectors that can bewritten in the form A~x for some vector~x.Theorem. Let A be an m by n matrix. The homogeneous systemA~x =~0 either has only the trivial solution~x =~0 or it has infinitelymany solns. Ifˆb 6=~0 is an m-vector, then the inhomogeneoussystemˆAˆx =ˆb has:1. No solns ifˆb is not in the range of A.2. Exactly one soln ifˆb is in the range of A and thehomogeneous system has only the trivial soln~x =~0.3. Infinitely many solns ifˆb is in the range of A and thehomogeneous system has a nontrivial soln~y 6=~0.5 / 16Range of a Matrix, No. of Solutions of a Linear SystemDefinition. The range of A is the set of all vectors that can bewritten in the form A~x for some vector~x.Theorem. Let A be an m by n matrix. The homogeneous systemA~x =~0 either has only the trivial solution~x =~0 or it has infinitelymany solns. Ifˆb 6=~0 is an m-vector, then the inhomogeneoussystemˆAˆx =ˆb has:1. No solns ifˆb is not in the range of A.2. Exactly one soln ifˆb is in the range of A and thehomogeneous system has only the trivial soln~x =~0.3. Infinitely many solns ifˆb is in the range of A and thehomogeneous system has a nontrivial soln~y 6=~0.5 / 16Another View of Matrix-Vector Multiplicationa11a12. . . a1na21a22. . . a2n.........am1am2. . . amnx1x2...xn=Pnj=1a1jxjPnj=1a2jxj...Pnj=1amjxj.Let~a1,~a2, . . .,~andenote the columns of A. Claim:A~x = x1~a1+ x2~a2+ . . . + xn~an,since this is justPnj=1a1jxjPnj=1a2jxj...Pnj=1amjxj.6 / 16Another View of Matrix-Vector Multiplicationa11a12. . . a1na21a22. . . a2n.........am1am2. . . amnx1x2...xn=Pnj=1a1jxjPnj=1a2jxj...Pnj=1amjxj.Let~a1,~a2, . . .,~andenote the columns of A. Claim:A~x = x1~a1+ x2~a2+ . . . + xn~an,since this is justPnj=1a1jxjPnj=1a2jxj...Pnj=1amjxj.6 / 16Another View of Matrix-Vector Multiplication, Cont.Example.1 2 34 5 67 8 9−121=1 · (−1) + 2 · 2 + 3 · 14 · (−1) + 5 · 2 + 6 · 17 · (−1) + 8 · 2 + 9 · 1=61218(−1) ·147+ 2 ·258+ 1 ·369=61218.7 / 16Linear Independence or Dependence of the Columns of ARecall that a set of vectors~a1, . . .~anare linearly independent if andonly if the only scalars c1, . . . , cnfor whichc1~a1+ . . . + cn~an=~0are c1= . . . = cn= 0.SinceA~x =nXj=1~ajxj,the homogeneous system A~x =~0 has only~x =~0 as solution if andonly if the columns~a1, . . . ,~anof A are linearly independent.8 / 16Linear Independence or Dependence of the Columns of ARecall that a set of vectors~a1, . . .~anare linearly independent if andonly if the only scalars c1, . . . , cnfor whichc1~a1+ . . . + cn~an=~0are c1= . . . = cn= 0.SinceA~x =nXj=1~ajxj,the homogeneous system A~x =~0 has only~x =~0 as solution if andonly if the columns~a1, . . . ,~anof A are linearly independent.8 / 16Nonsingular MatrixIf A is a square matrix (m = n), and if its columns are linearlyindependent, then they span Rn; i.e., every n-vectorˆb can bewritten as a linear combination of the columns of A; that isˆb liesin the range of A; that is, the equation Aˆx =ˆb has a solution, andsince A~x =~0 has only the trivial solution, the solution to Aˆx =ˆb isunique.Definition. An n by n matrix A is invertible or nonsingular if thereis an n by n matrix C such that AC = I . (Equivalently, there is ann by n matrix C such that CA = I .) The matrix C is denoted A−1.9 / 16Nonsingular MatrixIf A is a square matrix (m = n), and if its columns are linearlyindependent,

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