# You can only take the span of linearly independent vectors?

Ok, this might be a bit trivial but I'm having trouble wrapping my head around my text book.

So, to my understanding for ${Span(v_{1},v_{2},..,v_{n})}$ then ${v_{1},v_{2},..,v_{n}}$ must be linearly independent.

Is this correct or am I just completely misguided?

Thanks!

If you add a linearly dependent vector $v_{n+1}$ to the vector set, it is already in the $span \{v_{1}, ..., v_{n}\}$. So there is just no point in adding it. You can, but it just doesn't make sense to do so.
${Span(v_{1},v_{2},..,v_{n})}$ denotes the set of all linear combinations of vectors $v_1, ..., v_n$. Indeed, this makes perfect sense, and they need not be linearly independent. However, imagine {$v_1, ..., v_k$} $\subset$ {$v_1, ..., v_n$} is the largest possible subset of linearly independent vectors. Then we can conclude ${Span(v_{1},v_{2},..,v_{n})}$ = ${Span(v_{1},v_{2},..,v_{k})}$.