I know there's a theorem in linear algebra that has a list of statements that are equivalent, as follows:
- $\det(A) \ne 0$
- $Ax = 0$ has only the trivial solution
- $Ax = b$ is consistent for every matrix $b$
- $Ax = b$ has exactly one solution for every $n \times 1$ matrix $b$
(and several other which I do not need for the question that I'd like to ask).
So we've been learning about linear independence, span and bases, and I was wondering the following:
Is statement 2 the same as saying a set of vectors are linearly independent?
Is statement 3 the same as saying the set of vectors span a vector space? (Meaning that every single vector $b$ in the vector space can be made from some combination of $Ax$, where $A$ would represent that vectors and $x$ would be all of the scalar multiples of the vectors?
Is statement 4 the same as saying the set of vectors is a basis? (since that one solution must be the trivial solution, it must be linearly independent, and since it is consistent for every $b$, it spans the vector space?)
It seems pretty obvious that it's all related to each other, but I'm just wondering if they're all exactly the same.