Interpretation: Solution to a set of Linear Equations

[chaoseverlasting]

Hi,

While solving a system of linear equations, there are three possible cases - unique / infinite / no solutions - to the system.

One geometric interpretation is when one looks at a set of planes intersecting at one / many / no points respectively, for each of the above cases.

While going through what constitutes a vector space & how to check if a set of vectors belongs to a vector space (Sec. 7.4, q 27-36, p.302, Erwin Kreyzsig 9th Ed.), it seemed to me that the tests are similar to the Gauss Elimination method of checking for a single / many / no solutions via the rank of the augmented / coefficient matrices.

Is it possible to interpret the solutions (one, many, none) to a system of linear equations in terms of vector spaces?

I.e. if there is only one solution, the ranks of the augmented & coefficient matrices are equal - this means that we have identified a fundamental vector space for that system of equations.

If there are multiple solutions (infinitely many) to the system, then there is a more fundamental vector space for that system of equations and the augmented matrix / coeffiecient matrix vector space is a super set of that fundamental vector space - thus leading to infinitely many solutions.

For the no solution case, the vector spaces are dissimilar and unrelated, thus no solutions exist.

Apologies if the explanations are unclear - I'm still trying to understand the material better.

TIA for any insights / inputs.

 

[mfb]

Replace "vector spaces" by "sets of vectors" and it works. Every linear equation corresponds to a set of vectors satisfying this equation. If there is no solution, the intersection is empty. If there is one, the intersection has a single vector.. If there is an infinite set of solutions, you have a line/plane/... in the vector space as set of solutions.

 

[Othin]

Systems of linear equations are linear mappings, which are, in turn, a special class of functions. It is thus isomorphic to a function of n variables in, say, an Euclidean space (which we can visualize easier), and the usual considerations apply. If the system is completely satisfied, you have a point in this space. If you have one variable less than what you'd need, there's a degree of freedom, the variable is free to assume any values, in the same way a line would in an Euclidean space. And so on...