1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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


    User Avatar
    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook