onlinear1onlinear2ON SOLUTIONS OF LINEAR EQUATIONS Math 21b, O. KnillHomework Section 1.3: 4,14,34,48,50,26*,46*MATRIX. A re c tangular array of numbers is called a matrix.A =a11a12· · · a1na21a22· · · a2n· · · · · · · · · · · ·am1am2· · · amn=wwwwv v v12341 2 3A matrix with m rows and n columns is called a m × n matrix. A matrix with one column is a columnvector. The entries of a matrix are denoted aij, where i is the row number and j is the column number.ROW AND COLUMN PICTURE. Two interpretationsA~x =− ~w1−− ~w2−. . .− ~wm−|~x|=~w1· ~x~w2· ~x. . .~wm· ~xA~x =| | · · · |~v1~v2· · · ~vn| | · · · |x1x2· · ·xm= x1~v1+x2~v2+· · ·+xm~vm=~b .Row picture: each biis the dot product of a row vector ~wiwith ~x.Column picture:~b is a sum of scaled column vectors ~vj.”Row and Co lumn at Harvard”EXAMPLE. The system of linear equations3x − 4 y − 5z = 0−x + 2y − z = 0−x − y + 3z = 9is equivalent to A~x =~b, where A is a coefficientmatrix and ~x and~b ar e vectors.A =3 −4 −5−1 2 −1−1 −1 3, ~x =xyz,~b =009.The augmented matrix (separato rs for clarity)B =3 −4 −5 | 0−1 2 −1 | 0−1 −1 3 | 9.In this case, the row vectors o f A are~w1=3 −4 − 5 ~w2=−1 2 − 1 ~w3=−1 −1 3 The column vectors are~v1=3−1−1, ~v2=−4−2−1, ~v3=−5−13Row picture:0 = b1=3 −4 −5 ·xyzColumn picture:009= x13−1−1+ x23−1−1+ x33−1−1SOLUTIONS OF LINEAR EQUATIONS. A system A~x =~b with m equa-tions and n unknowns is defined by the m × n matrix A and the vector~b.The row reduced matrix rref(B) of B determines the number of solutionsof the system Ax = b. The rank rank(A) of a matrix A is the number ofleading ones in rref(A). There are three possibilities:• Consistent: Exactly one solution. There is a leading 1 in each columnof A but none in the last column of the augmented matrix B.• Inconsistent: No solutions. There is a le ading 1 in the last column ofthe augmented matrix B.• Consistent: Infinitely many solutions. There are columns of A with-out leading 1.If rank(A) = n, then there is exactly 1 solution.If rank(A) < rank(A|b),there are no solutions.If rank(A) = rank(A|b) < n: there are ∞ solutions.111111111111111(exactly one solution) (no solution) (infinitely many solutions)MURPHYS LAW.”If anything can go wrong, it will go wrong”.”If you ar e feeling good, don’t worry, you will get over it!””For Gauss-Jordan elimination, the error happens early inthe process and get unnoticed.MURPHYS LAW IS TRUE. Two equations could contradict each other. Geometrically, the two planes donot intersect. This is possible if they are parallel. Even without two planes b e ing parallel, it is pos sible thatthere is no intersection between all three of them. It is also possible that not enough equations are at hand orthat there are many solutions. Furthermore, there can be too ma ny eq uations and the planes do not intersect.RELEVANCE OF EXCEPTIONAL CASES. There a re impo rtant applications, where ”unusual” situationshappen: For example in medical tomography, systems of equations appear which are ”ill posed”. In this caseone has to be careful with the method.The linear equations are then obtained from a method called the Radontransform. The task for finding a goo d method had led to a Nobelprize in Medicis 1979 for Allan Cormack. Cormack had sabbaticals atHarvard and probably has done part of his work on tomography here.Tomography helps today for example for cancer treatment.MATRIX ALGEBRA. Matrices can be added, subtracted if they have the same s ize:A+B =a11a12· · · a1na21a22· · · a2n· · · · · · · · · · · ·am1am2· · · amn+b11b12· · · b1nb21b22· · · b2n· · · · · · · · · · · ·bm1bm2· · · bmn=a11+ b11a12+ b12· · · a1n+ b1na21+ b21a22+ b22· · · a2n+ b2n· · · · · · · · · · · ·am1+ bm2am2+ bm2· · · amn+ bmnThey can also be scaled by a scala r λ:λA = λa11a12· · · a1na21a22· · · a2n· · · · · · · · · · · ·am1am2· · · amn=λa11λa12· · · λa1nλa21λa22· · · λa2n· · · · · · · · · · · ·λam1λam2· · · λamnON SOLUTIONS OF LINEAR EQUATIONS Math 21b, O. KnillHomework Section 1.3: 4,14,34,48,50,26*,46*MATRIX. A re c tangular array of numbers is called a matrix.A =a11a12· · · a1na21a22· · · a2n· · · · · · · · · · · ·am1am2· · · amn=wwwwv v v12341 2 3A matrix with m rows and n columns is called a m × n matrix. A matrix with one column is a columnvector. The entries of a matrix are denoted aij, where i is the row number and j is the column number.ROW AND COLUMN PICTURE. Two interpretationsA~x =− ~w1−− ~w2−. . .− ~wm−|~x|=~w1· ~x~w2· ~x. . .~wm· ~xA~x =| | · · · |~v1~v2· · · ~vn| | · · · |x1x2· · ·xm= x1~v1+x2~v2+· · ·+xm~vm=~b .Row picture: each biis the dot product of a row vector ~wiwith ~x.Column picture:~b is a sum of scaled column vectors ~vj.”Row and Co lumn at Harvard”EXAMPLE. The system of linear equations3x − 4 y − 5z = 0−x + 2y − z = 0−x − y + 3z = 9is equivalent to A~x =~b, where A is a coefficientmatrix and ~x and~b ar e vectors.A =3 −4 −5−1 2 −1−1 −1 3, ~x =xyz,~b =009.The augmented matrix (separato rs for clarity)B =3 −4 −5 | 0−1 2 −1 | 0−1 −1 3 | 9.In this case, the row vectors o f A are~w1=3 −4 − 5 ~w2=−1 2 − 1 ~w3=−1 −1 3 The column vectors are~v1=3−1−1, ~v2=−4−2−1, ~v3=−5−13Row picture:0 = b1=3 −4 −5 ·xyzColumn picture:009= x13−1−1+ x23−1−1+ x33−1−1SOLUTIONS OF LINEAR EQUATIONS. A system A~x =~b with m equa-tions and n unknowns is defined by the m × n matrix A and the vector~b.The row reduced matrix rref(B) of B determines
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