I'm confused about whether or not a set $A$ of vectors in $\mathbb{R}^n$ can have less than $n$ vectors and be still be linearly independent. It would seem to me that to be linearly independent in $\mathbb{R}^n$ a set must have exactly $n$ vectors because otherwise taking each vector as a column and representing the set as a homogenous matrix in row echelon form you would end up with at least one free variable and thus infinite non-trivial solutions meaning that the set isn't linearly independent. ( Assuming that $|A|
Am I missing something here?