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