How to show a set of vectors does not span a vector space?

Let's say I am given a $4\times4$ matrix and I am to determine whether the columns of that matrix span $\mathbb R^4$. Please tell me if I'm correct:

One way to determine that is to calculate the matrix's determinant, if it is non-zero, then there is a unique solution and not only does it span R4, but it also is a basis for it, since it means that the column vectors are independent.

I'm assuming this is correct for now. But what if the determinant is equal to zero? From what I know, this does not imply the system is inconsistent, it might also be that there is an infinite number of solutions . If there is an infinite number of solutions, then the column vectors still span the vector space.(please tell me if that last sentence is valid as well)

If this is true, that would mean that my book, which only uses this method to determine whether a set of vector spans a vector space, assumes that that set actually spans the vector space and they don't think about the potential situation, in which the determinant is 0. And indeed, in every example they have given, they use that method and the result is that the determinant is 0, and so they conclude that the the set spans a vector space.

However, it would seem strange to me that they're using an inconclusive method, that is why I have doubts about my initial assumption that, if a determinant is not equal to 0, then the solutions are either infinite or they don't exist.

Could you please clarify this to me?

• in $\mathbb R ^n$, a set of n vectors spans the space if and only if the determinant of the matrix they form is nonzero. – Alan Dec 15 '14 at 13:29
• If the determinant is zero, it means that column vectors span a proper subspace $V$ of $R^4$ (not the whole space!). The existence of solutions in this case is equivalent to say that the right hand side if our system also belongs to $V$. Conversely, if for at least one right hand side there is an infinity of solutions, then the determinant is zero and the column vectors do not span the whole space. – TZakrevskiy Dec 15 '14 at 13:33