Given a collection of linearly independent vectors when can

Given a collection of linearly independent vectors, when can we add a vector and still have a linearly independent set? What would happen to that span?

Solution

If a vector is added to a linearly independent set of inearly dependent answer depends on the vector that is added. If the vector that is added isa linear combination of those in the collection, then the new collection is linearly dependent.

If the vector added is not in the span of the others, we get linear independenceof the new collection.