## What does it mean to not have a pivot in every row

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)?
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 Recognitions: Gold Member Science Advisor Staff Emeritus 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 R[sup]n[/itex] into a propersubspace of Rm so there exist b in Rm not in that subspace.