Register to reply

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

by Instinctlol
Tags: pivot
Share this thread:
Feb17-12, 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 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)?
Phys.Org News Partner Science news on
Experts defend operational earthquake forecasting, counter critiques
EU urged to convert TV frequencies to mobile broadband
Sierra Nevada freshwater runoff could drop 26 percent by 2100
Feb17-12, 05:21 PM
Sci Advisor
PF Gold
P: 39,568
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.

Register to reply

Related Discussions
Pivot Problem Introductory Physics Homework 6
What is a pivot column? Calculus & Beyond Homework 8
Rotating at a pivot Introductory Physics Homework 1
Friction of a pivot Mechanical Engineering 2
Force exerted by the floor on the feet of a student doing pushups Introductory Physics Homework 3