Problem Assume a finite set $F$, write the necessary and sufficient condition in terms of the number of elements of $F$, such that $F$ is a real vector space. (Assuming that the vector addition and scalar multiplication can be defined)
Attempt at Solution
Since the necessary conditions for being a vector space is to satisfy the 8 axioms, I thought that $F$ will require at least 3 elements, an identity element, an arbitrary element and the inverse of it.
How could I go from here?
Best Answer
Since a real vector space of positive dimension necessarily contains infinitely many elements, $F$ must be the zero vector space. This suggests that it is necessary and sufficient that $F$ be a singleton set.
Depending on the intention of the problem, there's probably still a couple things to be proven regarding the operations of addition and scalar multiplication, namely that there is only one binary operation on a singleton set (which gives addition), and there is only one possible map $\mathbb R \times F \to F$ giving scalar multiplication.