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 ?
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1K is mapping, I mean K maps set into V, see this – xrfxlp Jul 02 '19 at 00:54
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What do you think a vector space is without a field? – anomaly Jul 02 '19 at 01:03
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A field is an algebraic structure where there's a set with addition and multiplication satisfying certain properties (such as distributive, associative, commutative). Classic examples are $\mathbb R$ and $\mathbb C$, but there are also finite fields, such as integers modulo a prime. – J. W. Tanner Jul 02 '19 at 01:06
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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 $2$.
Ethan Bolker
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