Understanding Defiinition of Vector Space

Question

Let $F$ be a field. A vector space over $F$ is a set $V$ together with $+$,$\cdot$ satisfiyng:

$$+: V \times V \rightarrow V$$ $$\cdot: F \times V \rightarrow V$$

with usual properties.

My question is that why did not we write as a $$+: F \times V \rightarrow V$$

Answer 1

That would be a different kind of structure. The idea of a vector space is to have a "parallelogram" type construct that permits the addition of two vectors to obtain a third vector (as well as the scaling of a given vector by a scale factor from $F,$ that is what the scalar multiplication is about).

Answer 2

Writing $+: F \times V \rightarrow V$ would mean that $+$ takes an element from the field $F$ and another element from the vector space $V$. This is NOT the same as $+:V \times V \rightarrow V$.

Unless you're asking why we don't have two operations taking $F \times V$ to $V$. In which case, we would get a completely different structure to what we want as Justpassingby has already pointed out.

Answer 3

$+:F \times V \to V$ would mean that you can add a number to a vector to get a vector. Addition, as we'd like to think about it in a vector space, takes two vectors and gives you another.