Can a 3x5 Matrix with 3 Free Variables Ever Have No Solution for Any Vector b?

[sami23]

Homework Statement

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.

Homework Equations

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?

 

[deluks917]

This doesn't make sense. Most systems of 3 equations in 5 unknowns have many solutions.

 

[Ray Vickson]

Homework Statement

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.

Homework Equations

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?

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

RGV