1. The problem statement, all variables and given/known data Suppose B and A are a basis in vector space V, dimension N, now if I can any linear independent columns of basis B and A can I combine them to make a new basis C provided that all the all, even the unused vectors of bases A & B linear combinations of C. 2. Relevant equations 3. The attempt at a solution I guess the key thing to do here would be to look at the null space of C and see how it compares to A, if this is a problem in R^n then if the null spaces where the same then so would be the basis. So I guess I am just looking for a general method of figuring out if the span of two column spaces are the same, is the question posed true or false?