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In many books and on Wikipedia a vector space is defined as a tuple $(F, +, V)$ where $F$ is a field and $V$ an abelian Group plus some axiums that must hold which I will omit here.

I also often see the phrase 'a vector space over $F$'.

What I am missing in all definitions that I have come across it the relationship between $F$ and $V$.

So my question is this:

How are the elements of $V$ related to $F$? Is it a general requirement that $V=F^n$? That is does each component $x_i$ of a vector $x \in V$ with $i \in n$ have to satisfy $x_i \in F$? Obviously thats the case for the common real and complex vector spaces $\mathbb R^n$ and $\mathbb C^n$, but is it also that way in the general case?

If not, can you give an example of a vector space where this not the case?

noctusraid
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lanoxx
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1 Answers1

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The definition of the field $F$ is essential in defining the properties of the vector space. The same set $V$ of vectors, is a different vector space over different fields $F$.

As a dramatic example, consider $\mathbb{R}$ as the set of vectors, with the usual addition, than we can think at $\mathbb{R}$ as a vector space over $\mathbb{R}$ and in this case it has dimension $1$. But, it can be also a vecor space over the field $\mathbb{Q}$ of rational numbers, and in this case it is a vector space of uncountable infinite dimension. (See Is there a quick proof as to why the vector space of $\mathbb{R}$ over $\mathbb{Q}$ is infinite-dimensional?).

Another, less sophisticated example is $\mathbb{C}$. It is a vector space of dimension $1$ over $\mathbb{C}$ and a vector space of dimension $2$ over $\mathbb{R}$ (and it has uncountable infinite dimension as a vector space over $\mathbb{Q}$).

About the last question: note that the components of a vector are defined with respect to some basis and are the coefficients of a linear combination, so they are elements of the field.

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Emilio Novati
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  • So I could have a vector space $(\mathbb Q, +, \mathbb R^n)$, but then according to your link $n$ would have to be infinite; thus $(\mathbb Q, +, \mathbb R^\infty)$. And in the case of the complex numbers I could write: $(\mathbb C, +, \mathbb C^1)$ and $(\mathbb R, +, \mathbb C^2)$. Is this the correct terminology? – lanoxx Jan 26 '16 at 13:49
  • No. $(\mathbb{Q},+,\mathbb{R}^n)$ has infinite dimension also if $n=1$. In other words, we cannot say what is the dimension of $(\mathbb{K},+,\mathbb{R}^n)$ if the field $\mathbb{K}$ is not specified. If $\mathbb{K}=\mathbb{R}$ than the dimension is $n$, if $\mathbb{K}=\mathbb{Q}$ the dimension is infinite for all $n>0$. – Emilio Novati Jan 26 '16 at 15:30
  • For the complex case. $(\mathbb{C},+,\mathbb{C}^n)$ has dimension $n$, but $(\mathbb{R},+,\mathbb{C}^n)$ has dimension $2n$. So, $(\mathbb{R},+,\mathbb{C}^2)$ has dimension $4$. – Emilio Novati Jan 26 '16 at 15:32