# Linear algebra basis question

## 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 $V$?

## 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
Homework Helper
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.

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