Vector spaces need to fulfill a set of axioms to be labeled as such. These include the requisite of the zero vector being included in the set.
However, the usual way to reason about matrices hinges upon the concept of column and row spaces.
Since the immediate intuition (and definition) is the span of the column (or row) vectors, the working idea is clear. Yet, it would seem as though accepting that any old real-valued matrix will necessarily carry with it a column space is tantamount to saying than any two vectors (or one single vector) forms a space.
So, is it true that any set of vectors packed into a real-valued matrix is necessarily the basis of a vector space? If so, how can we see that they meet the criteria (for example, the inclusion of a zero additive element)?