I think I have something mixed up so if someone can please point out my error. 1. the set of all linear combinations is called a span. 2. If a family of vectors is linearly independent none of them can be written as a linear combination of finitely many other vectors in the collection. 3. If the determinant of a matrix is not equal to zero the vectors are linearly independent. Therefore, if the determinant of the matrix does not equal zero the vectors can not be written as a linear combination, hence there is no span.