Question

For the span of a set of vectors, there are 2 methods to prove they span a certain vector space:
1) if det is non zero.
2) if the solution is consistent; what do they mean by consistent? (Unique AND infinite many solutions) or just unique?

Answer 1

there's something odd about the question. if the nxn matrix is invertible, then it spans \(\mathcal{R}^n\) and will also have one unique solution \(\mathbf x = A^{-1} \mathbf b\)

but even if it's singular, it can still span \(\mathcal{R}^r\), \(r < n\) depending on how many independent vectors it comprises.

it's all In Strang's list i posted on other thread, though maybe you have to eek it out a bit

see here for how consistent has a very broad meaning.....so it would seem

https://en.wikipedia.org/wiki/Consistent_and_inconsistent_equations