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

[Instinctlol]

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 R^m, the equation Ax has a solution
2. Each b in R^m is a linear combination of the columns of A.
3. The columns of A span R^m
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)?

 

[HallsofIvy]

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^{n[/itex] into a propersubspace of Rm so there exist b in Rm not in that subspace.}