Do These Vectors Form a Basis for the Vector Space?

[jimmycricket]

Homework Statement

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 $V$?

Homework Equations

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?

 

[HallsofIvy]

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.

 

[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.

 

[Office_Shredder]

If $v_1,...,v_k$ span $V$ but $v_1,...,v_{k-1}$ do not then $v_1,...,v_k$ are linearly independent.

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)