4.5 Row and Column Spaces

In numerous examples we have observed the phenomenon of “disappearing equations” that sometimes occurs when we solve a linear system using the method of Gaussian elimination. The appearance in this process of a trivial equation 0=0 means that one of the original equations was redundant. For instance, in the system


the third equation provides no additional information about a solution (x, y, z) because it is merely the sum of the first two equations.

Given a homogeneous linear system, it is natural to ask how many of the equations are “irredundant,” and which ones they are. We will see that an answer to this question leads to a natural and simple relation between the number of irredundant ...

Get Differential Equations and Linear Algebra, 4th Edition now with the O’Reilly learning platform.

O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.