Linearly Independent Vectors: Deleting a Vector & Its Impact

[Dustinsfl]

Let x₁, x₂,...,x_k be linear independent vectors in a vector space V.

If we delete a vector, say x_k, from the collection, will we still have a linearly independent collection of vectors? Explain.

By deleting a vector from linearly independent span, the other vectors will remain independent; however, I don't know how to prove it.

 

[snipez90]

So delete one, say v_i. If the remaining vectors are linearly dependent, then there exist scalars for which the linear combination of the remaining vectors equals 0. What happens when you add 0*(v_i) to both sides?