Let set of vectors $\{x,y,z\}$ be linearly independent. Then would $\{x,y,z,x\}=\{x,y,z\}$ be linearly dependent, also?
If so, that seems like a problem (since $\alpha x+\beta y+\gamma z+(-\alpha)x=0$ would allow for a non-zero $\alpha$) unless it is understood that each vector in a set of vectors is used only once in a linear combination of that set.
I've seen one author use tuples to get around this. Is that the way to go?