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Part I Suppose we wish to solve the system of equations x + y + z -3 2x + 3y + z 3x + 4y + 2z – 12 The augmented matrix for this system is 1 1 1 -3 Reducing to echelon form gives us 0 1 1 - 3 from which - | 0 0 we can see that column 3 is a free column. If we set the third variable, z, equal to 0 and then use back substitution, we get y and then x = Therefore the particular solution to the system Ax bis xp Also, considering the nullspace problem Ax solution xn = 0, we get the special 00

Part I Suppose we wish to solve the system of equations x + y + z -3 2x + 3y + z 3x + 4y + 2z – 12 The augmented matrix for this system is 1 1 1 -3 Reducing to echelon form gives us 0 1 1 - 3 from which - | 0 0 we can see that column 3 is a free column. If we set the third variable, z, equal to 0 and then use back substitution, we get y and then x = Therefore the particular solution to the system Ax bis xp Also, considering the nullspace problem Ax solution xn = 0, we get the special 00

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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Part I Suppose we wish to solve the system of equations x + y + Z = - 3 2x + 3y + z = – 9 Зх + 4у + 2х — - 12 - The augmented matrix for this system is 1 1 1 - 3 Reducing to echelon form gives us 0 1 - 1 - 3 from which 0 0 we can see that column 3 is a free column. If we set the third variable, z, equal to 0 and then use back substitution, we get y = and then x = Therefore the particular solution to the system Ax = b is ap Also, considering the nullspace problem Ax solution xn 0, we get the special The general solution of Ax x = xp + cxn. b is the set of all vectors of the form
Transcribed Image Text:Part I Suppose we wish to solve the system of equations x + y + Z = - 3 2x + 3y + z = – 9 Зх + 4у + 2х — - 12 - The augmented matrix for this system is 1 1 1 - 3 Reducing to echelon form gives us 0 1 - 1 - 3 from which 0 0 we can see that column 3 is a free column. If we set the third variable, z, equal to 0 and then use back substitution, we get y = and then x = Therefore the particular solution to the system Ax = b is ap Also, considering the nullspace problem Ax solution xn 0, we get the special The general solution of Ax x = xp + cxn. b is the set of all vectors of the form
Part II Suppose that B is a 4x3 matrix with full column rank. Then the number of pivot columns is and the number of free variables is In this case, the only vector in the nullspace of Bis The general problem Bx = d then has at most solution(s). If C is a 2x4 matrix with full row rank, then the number of special solutions for the nullspace problem Cx = 0 is O is and therefore, if C'x d has a solution, then it actually has infinitely many.
Transcribed Image Text:Part II Suppose that B is a 4x3 matrix with full column rank. Then the number of pivot columns is and the number of free variables is In this case, the only vector in the nullspace of Bis The general problem Bx = d then has at most solution(s). If C is a 2x4 matrix with full row rank, then the number of special solutions for the nullspace problem Cx = 0 is O is and therefore, if C'x d has a solution, then it actually has infinitely many.
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