You can only take the span of linearly independent vectors?

Question

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!

Answer 1

No, you can take the span of any set of vectors. In order for a set to be a basis of a subspace, it must not only span the subspace, but also be linearly independent.

The span of any set of vectors is simply the collection of all linear combinations of the vectors in that set. That's defined for any set.

Answer 2

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.

Answer 3

${Span(v_{1},v_{2},..,v_{n})}$ 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 ${Span(v_{1},v_{2},..,v_{n})}$ = ${Span(v_{1},v_{2},..,v_{k})}$.