Any linearly independent set in a vector space is a basis for that space? Is that true or false in general?
I would think it would be true because the fact that is it a linearly independent set would force the set of contain a pivot position in every row in echelon form. Since it has a pivot pos in every row, the set can span the space.
Best Answer
If a set of vectors is linearly independent, any of its subsets is, too. But whenever you remove an element from a linearly independent set, the removed vector is not in the span of the remaining ones.
So, removing an element from a basis gives a linearly independent set which is not a basis.