Confusion in the definition of Vector space

Question

One of the axiom while defining vector space $(V, +, .)$ says that $1.v = v$, where $v\in V$. I am confused why this axiom is needed when we already know that for set $V$ to be a vector space over some field say $F$, scalar multiplication should hold? What is the significance of this axiom? How this axiom ($1.v = v$) is different from the scalar multiplication axiom of vector space? Is the element $1$ is the identity element of the field $F$ with respect to operation multiplication?

Kindly help me. Thank you

Answer 1

The definition of scalar multiplication says that $1\cdot v$ is a vector in $V$, but it does not say what that vector is equal to. It is an axiom because this property is a desirable property and it is not derivable from the other axioms.