Adding a Vector to a Spanning Set: Will it Still Span the Vector Space?

[Roni1985]

Homework Statement

Let {x1,x2,...,x_k} be a spanning set for a vector space V.
a) if we add another vector, x_k+1 to the set, will we still have a spanning set ? Explain

Homework Equations

The Attempt at a Solution

I think 'yes', but I am not sure if my explanation is correct.
It doesn't matter if they are independent or dependent, because we know that the first k vectors are independent and that's what we need to create a spanning set for V.
If the kth+1 vector is independent, the set is going to span a bigger vector space, a vector space that includes V. So, it's going to be a spanning set for V regardless of x_k+1

Is my reasoning correct ?

Thanks.

 

[radou]

Yes, it doesn't matter if they're independent or not, the set which contains xk+1 still spans V. And you don't know if the first k vectors are independent, as a matter of fact. Suppose dimV < k, for example.