Empty Set $\{\}$ is the Only Basis of the Zero Vector Space $\{0\}$ Question
Suppose we want to find a basis for the vector space $\{0\}$. 
I know that the answer is that the only basis is the empty set.
Is this answer a definition itself or it is a result of the definitions for linearly independent/dependent sets and Spanning/Generating sets? If it is a result then would you mind mentioning the definitions of bold items which based on them this answer can be deduced.

Useful Links
These are the links that I found useful for answering this question. It needs some elementary background form mathematical logic. You can learn it by spending a few hours on this Wikipedia page.
Link 1, Link 2, Link 3, Link 4, Link 5, Link 6

The standard definition of basis in vector spaces is:

$\mathcal B$ is a basis of a space $X$ if:


*

*$\mathcal B$ is linearly independent.

*The span of $\mathcal B$ is $X$.



You can easily show both of these statements are true when $X=\{0\}$ and $\mathcal B= \{\}$. Again, you have to look at the definitions:


*

*Is $\{\}$ linearly independent? Well, a set $A$ is linearly independent if, for every nonempty finite subset $\{a_1,a_2\dots, a_n\}$, we have that if $$\alpha_1a_1 + \dots + \alpha_n a_n=0,$$ then $\alpha_i=0$ for all $i$. This condition is satisfied automaticall in the case of an empty set (everything follows from a false statement). This part may be difficult to understand, but since there is no nonempty finite collection of vectors from $\{\}$, any statement you say about nonempty finite collections of vectors from $\{\}$ must be true (because any such statement includes an assumption that a nonempty finite collection exists. It does not, meaning that any such statement is of the type $F\to A$ and is automatically true). This means $\{\}$ is linearly independent.

*Is the span of $\{\}$ equal to $\{0\}$? Well, the span of a set $A\subseteq X$ is defined as the smallest vector subspace of $X$ that contains $A$. Since all vector subspaces contain $\{\}$, it is clear that $\{0\}$, which is the smallest vector subspace at all, must be the span of $\{\}$.

Alternatively, the span of $A$ is the intersection of all vector subspaces that contain $A$. Again, it should be obvious that this implies that the span of $\{\}$ is $\{0\}$.
