"V is a vector space over the field K"

Question

I can visualize what can a vector space be, but I am not able to comprehend exactly what is the field K here. Can someone explain in basic terms. (or) What exactly is a field here ?

Answer 1

Not a formal definition, but perhaps something that will help your intuition.

The $n$ dimensional vector space you are most used to is the set of $n$-tuples $(a_1, a_2, \ldots, a_n)$ where each $a_i$ is a real number (an element of the field $\mathbb{R}$).

All the rules for vectors work just as well if you restrict the coordinates to be rational numbers (the field $\mathbb{Q}$), or allow them to be complex numbers (the field $\mathbb{C}$).

A field is just a set where the ordinary rules of arithmetic work. Any field will do for the coordinates for vectors. For example, they can just be $0$ or $1$, with arithmetic modulo $2$.