Patterns of Solution Sets of a System of Linear Equations

In summary, the author describes the pattern of solutions for a system of linear equations as having a particular solution added to an unrestricted combination of other vectors. He also mentions that a zero-element solution set, although fitting the pattern, has no particular solution. However, this may be a special case and not worth worrying about, as the use of language is inconsequential. It resembles solving non-homogeneous equations and may involve finding the general solution and adding a single solution to the entire equation.
  • #1
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I'm reading through the book "Linear Algebra", by Jim Hefferon (which you can download for free!). In section I.3, he describes that the pattern of solutions for a system of linear equations:

"They have a vector that is a particular solution of the system added to an unrestrictred combination of some other vectors."

Then he goes on to say:

"A zero-element solution set fits the pattern since there is no particular solution, and so the set of sums of that form is empty."

Isn't he contradicting himself here? First, he says the pattern has a vector of a particular solution, and then he says a zero-element solution fits the pattern because it has no particular solution! Can someone clarify this?
 
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  • #2
well maybe he is overly optimistic at trying to describe all situations in the same language.

there are two kinds of systems, those with solutions and those without.

if a system AX=b has solutions, then the difference of any two solutions is a solution of the homogeneous system AX=0.

conversely given one particular solution of AX=b, every other solution can be obtained by adding to that one, all solutions of the system AX=0.


so if there are no solutions, then i would be challenged trying to claim that is a special case of this situation. such things as his use of language are really inconsequential and therefore not worth worrying about, in my view.
 
  • #3
Sounds a lot like solving non-homogeneous equation. I suspect that he is looking at equations of the form [tex]Ax= \lambda x+ c[/tex]. If [tex]\lambda[/tex] is NOT an eigenvalue of A, that equation has only one solution. If [tex]\lambda[/tex] IS an eigenvalue of A then it has either no or an infinite number of solutions (the "Fredholm alternative"). Find the "general" solution to [tex]Ax= \lambda x[/tex] (the eigenvectors) and add a single solution to the entire equation (if there is one) to get the "general" solution to entire equation.
 

Related to Patterns of Solution Sets of a System of Linear Equations

What are "Patterns of Solution Sets of a System of Linear Equations"?

Patterns of Solution Sets of a System of Linear Equations refer to the different ways in which a system of linear equations can be solved. These patterns can be identified by analyzing the coefficients and constants of the equations and determining the number of solutions and the nature of those solutions.

What are the different types of solution sets in a system of linear equations?

The three main types of solution sets in a system of linear equations are consistent, inconsistent, and dependent. A consistent system has at least one solution, an inconsistent system has no solutions, and a dependent system has infinitely many solutions.

How can I determine the number of solutions in a system of linear equations?

The number of solutions in a system of linear equations can be determined by analyzing the coefficients and constants of the equations using methods such as substitution or elimination. The resulting equations will reveal whether the system is consistent, inconsistent, or dependent, and thus the number of solutions.

What is the difference between a consistent and an inconsistent system of linear equations?

A consistent system of linear equations has at least one solution, meaning the equations intersect at a single point. An inconsistent system has no solutions, meaning the equations do not intersect and are parallel to each other. In other words, the equations in a consistent system are solvable, while those in an inconsistent system are not.

Why is it important to understand the patterns of solution sets in a system of linear equations?

Understanding the patterns of solution sets in a system of linear equations is important because it allows us to determine the nature of the solutions and whether a solution exists at all. This knowledge is crucial in solving real-world problems that can be modeled using linear equations, such as in engineering, economics, and science.

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