Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

  1. Feb 17, 2012 #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. Feb 17, 2012 #2

    HallsofIvy

    User Avatar
    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Similar Discussions: What does it mean to not have a pivot in every row
  1. What does X'X mean ? (Replies: 4)

Loading...