Patterns of Solution Sets of a System of Linear Equations

[e(ho0n3]

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?

 

[mathwonk]

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.

 

[HallsofIvy]

Sounds a lot like solving non-homogeneous equation. I suspect that he is looking at equations of the form $$Ax= \lambda x+ c$$. If $$\lambda$$ is NOT an eigenvalue of A, that equation has only one solution. If $$\lambda$$ IS an eigenvalue of A then it has either no or an infinite number of solutions (the "Fredholm alternative"). Find the "general" solution to $$Ax= \lambda x$$ (the eigenvectors) and add a single solution to the entire equation (if there is one) to get the "general" solution to entire equation.