I am getting a bit confused with the terminology here. I understand that a field means some set of scalars like real numbers but why do we need a field for a vector space? Are not the numerical values used to define a vector are inherent properties of the vector space? Why do we term it like "vector space over real field"? If not numbers, what are the other fields possible for a vector space because you obviously numbers to define the value of magnitude and direction of the vectors?
[Math] Why do we call a vector space in terms of vector space over some field
linear algebraterminology
Best Answer
A vector space is defined as a quadruple $(V,K,+,\cdot)$ where $V$ is a set whose elements are called ''vectors'' , $K$ is a field, $+\colon V\times V \rightarrow V$ and $\cdot\colon K\times V \rightarrow V$ are operations that satisfy a suitable set of axioms (see here).
For the same set $V$ we can define different vector spaces changing the field $F$ and the difference can be dramatic.
The simpler case is for $V=\mathbb{R}$. If we take $K=\mathbb{R}$ with the usual addition and multiplication, we have a vector space of dimension $1$, but if we take $K=\mathbb{Q}$ we have a vector space that has a uncountable basis so its dimension is infinite (Is there a quick proof as to why the vector space of $\mathbb{R}$ over $\mathbb{Q}$ is infinite-dimensional?).
Another, less dramatic, example, is if $V=\mathbb{C}^n$: if we choose $K= \mathbb{C}$ than we have a vector space (with the usual operations) over $\mathbb{C}$ of dimension $n$, if we choose $K= \mathbb{R}$ we can obtain a space of dimension $2n$.
Again, if $K=\mathbb{Q}$ the change is more relevant and we have a space of infinite (not countable) dimension.