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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

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

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