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What does it mean to not have a pivot in every rowby Instinctlol
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#1
Feb1712, 02:47 PM

P: 79

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)? 


#2
Feb1712, 05:21 PM

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P: 39,303

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 R^{m} so there exist b in R^{m} not in that subspace. 


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