Systems of Linear Equations Underdetermined and Overdetermined systems

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Systems of Linear Equations: Underdetermined and Overdetermined systems Theorem 1. If the number of

Systems of Linear Equations: Underdetermined and Overdetermined systems Theorem 1. If the number of equations is greater than or equal to the number of variables then the system has no solution, one solution, or infinitely many solutions. 2. If the number of equations is less than the number of variables, then the system has no solution or infinitely many solutions. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Ex. A system with no solution: Matrix The system is inconsistent and has NO

Ex. A system with no solution: Matrix The system is inconsistent and has NO solution. Notice the false statement 0 = 1 Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Ex. A system with infinitely many solutions: Matrix Notice the row of zeros. .

Ex. A system with infinitely many solutions: Matrix Notice the row of zeros. . Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Example (Cont. ) So or If we let z = t then the solution

Example (Cont. ) So or If we let z = t then the solution is given by (2 – t, 1 – t, t) Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Ex. A system with more equations than variables: Matrix No Solution. Notice the false

Ex. A system with more equations than variables: Matrix No Solution. Notice the false statement 0 = 1. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Ex. A system with more variables than equations: Matrix So or Infinitely many solutions.

Ex. A system with more variables than equations: Matrix So or Infinitely many solutions. If we let z = s and w = t then the solution is given by (1 + 2 s + t, s, t) . . . Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.