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)