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!

Linear Algebra - Rank Theorem

  1. Nov 4, 2011 #1
    1. The problem statement, all variables and given/known data
    Suppose a non-homogeneous system, Ax = b, of 3 linear equations in 5 unknowns (3x5 matrix) and 3 free variables, prove there is no solution for any vector b.


    2. Relevant equations
    Using the rank theroem:
    n = rank A + dim Nul(A) where n = # of columns; dim Nul(A) = # free variables


    3. The attempt at a solution
    rank A = n - dim Nul(A) = 5-3 = 2 (which represents the pivot columns)

    How do I know there are no solutions for any vector b knowing there can be at most 3 pivot columns?
     
  2. jcsd
  3. Nov 4, 2011 #2
    This doesn't make sense. Most systems of 3 equations in 5 unknowns have many solutions.
     
  4. Nov 5, 2011 #3

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper


    If the matrix has rank 3 there are infinitely many solutions.

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




Similar Discussions: Linear Algebra - Rank Theorem
  1. Algebra Theorem (Replies: 11)

  2. Linear Algebra (Replies: 1)

Loading...