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.