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What does it mean to not have a pivot in every row

  1. I am trying to fully understand this theorem

    Theorem: Let A be an m x n matrix. The following are all true or all false.
    1. For each b in Rm, the equation Ax has a solution
    2. Each b in Rm is a linear combination of the columns of A.
    3. The columns of A span Rm
    4. A has a pivot position in every row.

    So when A does not have a pivot in every row, it disproves (1) because each b will not have a solution.

    How would you disprove (2) with (4)?
     
  2. jcsd
  3. HallsofIvy

    HallsofIvy 40,410
    Staff Emeritus
    Science Advisor

    If A does not have a pivot in every row, then its determinant is 0 and A is not invertible.

    When you say "disprove (2) with (4)" do you mean disprove (2) assuming (4) is NOT true?

    If A does not have a pivot in every row, then A maps Rn[/itex] into a propersubspace of Rm so there exist b in Rm not in that subspace.
     
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