Linear algebra basis question

1. Jan 8, 2014

jimmycricket

1. The problem statement, all variables and given/known data
Let $v_1,...,v_k$ be vectors in a vector space $V$. If $v_1,...,v_k$ span $V$ and after removing any of the vectors the remaining $k-1$ vectors do not span $V$ then $v_1,...,v_k$ is a basis of $V$?

2. Relevant equations

3. The attempt at a solution
If $v_1,...,v_k$ span $V$ but $v_1,...,v_{k-1}$ do not then $v_1,...,v_k$ are linearly independent.
If $v_1,...,v_k$ span $V$ and are linearly independent the $v_1,...,v_k$ is a basis of $V$
Is this reasoning correct?

2. Jan 8, 2014

HallsofIvy

Staff Emeritus
Yes, your reasoning is correct. If any subset of this set of vectors does not span the vector space, then the original set is independent.

3. Jan 8, 2014

1MileCrash

If I were writing a proof I would want to emphasize that $v_{k}$ is any arbitrary vector of the set and not a named one.

Personally I would say:
{${v_{1}, v_{2}, ... v_{k}}$} \ {${v_{i}}$} is linearly dependent for all i in {1,2,..,k}.

But I'm just being nitpicky.

4. Jan 8, 2014

Office_Shredder

Staff Emeritus
This isn't a mathematical point but given the level of the exercise I would guess you are expected to prove this part (but obviously you are the only one who can know what level of detail is required in your homework)