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I really need help understanding this sentence right here:

"Every vector space is regarded as a vector space over a given field, which is denoted by F."

I know from my professor and the book that: 1) Vector space= a set of vectors in which 2 operations are defined ( addition and scalar multiplication) so that when you add any 2 vectors in the vector space, that unique element would also be in the vector space, and the same holds for scalar multiplication.

2)Field=the set of all real numbers or complex numbers.

My question is: what does it mean to have a vector space over a field?

Thanks you so much.

user6873843
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  • Have you see the wikipedia page for this? We have scalar multiples $\lambda v$ for vectors $v\in V$, with $\lambda\in F$. – Dietrich Burde Sep 24 '16 at 20:32
  • It means that the set of numbers $a$ you use for the operation $a$ times $V$ has the properties of a field, in particular is $\mathbb{R}$ (a structure that may be you haven't been introduced enough. It would be a catastrophe if instead of the field $\mathbb{R}$ we had the ring $\mathbb{Z}$, because for example $U=2V$ couldn't be inverted into $V=\frac{1}{2}U$... – Jean Marie Sep 24 '16 at 20:33

2 Answers2

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You mention scalar multiplication, but what are you multiplying by? The set of elements we can multiply by is a field, which is what they are calling $F$. Every vector space must have a specified field that you take scalars from

Alex Mathers
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  • I am not certain that you address the real question the OP is asking. – Jean Marie Sep 24 '16 at 20:36
  • @JeanMarie you don't think so? It seems that OP knows what a vector space is but confused by the terminology "vector space over $F$"; I am just trying to clarify that this is just a more specific way of stating that $V$ is a vector space where the scalars come from $F$. – Alex Mathers Sep 24 '16 at 20:39
  • In fact, maybe, you are right. I thought at first that the OP was bothered by the word "field" (instead of ring, etc...), but hopefully, this is not the case. – Jean Marie Sep 24 '16 at 20:44
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First of all, you have some problems with the definitions you use.

Don't think of a vector space as "a set of vectors". This is circular and requires you to define a vector first!

A vector space is a set of elements which satisfies the vector space axioms, and these are more expansive than just closure under the operations.

A field need not be the reals or the complex, either. The rationals form a field, and so do the integers modulo a prime. Fields are a requirement because, like vector spaces, they are a closed space under the operations.

In a way, vector spaces are a generalisation of fields to multiple dimensions; you can consider fields to be a vector space where all of the vectors are one-dimensional. I suggest some exploration of this, to help you get comfortable with seeing the connection and how properties from one might be similar or different in the other.

Elements of the field are used to make up elements of the vector space (which are called vectors). The use of a field ensures that when you employ scalar multiplication or linear combination, the field elements are guaranteed to give you a result also in that field. Therefore the vector space elements, i.e. vectors, will be composed of field elements, so they will also be in the vector space!

Nij
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