Wikipedia:Reference desk/Archives/Mathematics/2011 November 26

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November 26

Vector Spaces Over Finite Fields

Reading about error detection and correction codes, I came across finite vector spaces and now I am a bit confused. If I consider a finite field, let's say $\mathbb {F} _{2}$, then a coordinate space like $\mathbb {F} _{2}^{10}$ is a finite vector space with dimension ten. But does it make sense to say for example that the real numbers can be considered a vector space over $\mathbb {F} _{2}$? As I understand my undergrad linear algebra class, the only requirement was that the scalars come from a field. It doesn't really matter what the supposed vector space is. Let's say I even define multiplication between a scalar from $\mathbb {F} _{2}$ and a vector from $\mathbb {R}$ as standard real multiplication. Do I have a valid vector space? What would be its basis and what dimension would it be? And if this all makes sense, then can I do the same with function spaces? Can I consider the space of all polynomials from reals to reals as a vector space over $\mathbb {F} _{2}$ or does it have to be polynomials from $\mathbb {F} _{2}$ to $\mathbb {F} _{2}$ to be considered a vector space over $\mathbb {F} _{2}$?

Incidentally, are there any good introductory books on finite vector spaces anyone can recommend? Kind of like undergrad introductory linear algebra books, with the same kind of development of linear algebra but over finite fields instead of reals or complex? Every time I search for one, I keep getting books on "finite-dimensional" vector spaces. I would like to learn more and explore the subtleties of finite vector spaces with issues like inner products, bases, dimensions, orthogonality, etc. Thanks.65.101.251.155 (talk) 00:33, 26 November 2011 (UTC)

When you say you'd define multiplication between a ∈ F₂ and x ∈ R as the ordinary product ax, I assume you are identifying a with one of the real numbers 0 and 1. In that case, your multiplication will fail to satisfy the distributive law, since (1 + 1)⋅3 = 0⋅3 = 0, but 1⋅3 + 1⋅3 = 3 + 3 = 6. Generally, given an addiitive group G, it is possible to turn it into an F₂-vector space if and only if every x ∈ G satisfies x + x = 0. On the other hand, if K is a field which is a subring of a ring R, then R is a K-vector space with exactly the multiplication you defined. The problem here is that F₂ is not a subfield of R. 96.46.192.31 (talk) 06:53, 26 November 2011 (UTC)

So does this mean that for any given vector field (except for the trivial vector space containing only the zero element), the underlying field can only be something which is isomorphic to a field contained in the vector space itself? So $\mathbb {R} ^{3}$ can be a vector space over $\mathbb {R}$ or even $\mathbb {Q}$ but it cannot be a vector space over $\mathbb {C}$ nor over $\mathbb {F} _{2}$? $\mathbb {C} ^{3}$ can be a vector space over $\mathbb {C}$ or $\mathbb {R}$ or $\mathbb {Q}$? Similarly $\mathbb {F} _{8}^{10}$ can be considered a vector space over $\mathbb {F} _{8}$ or $\mathbb {F} _{2}$ but not over $\mathbb {R}$ or $\mathbb {F} _{4}$ because those are not "subfields"? Also, can anyone recommend any good books for this subject? Thanks. 65.101.251.155 (talk) 04:05, 29 November 2011 (UTC)

By "vector field," I assume you mean "field which is a vector space." All of these examples are correct, if what you mean by "can be considered as a vector space over" such and such field means "can be considered in an obvious way." In some of these cases, there exist vector space structures, but they are not defined in an obvious way.

Here is an example in which it can be done though. R⁴ is obviously isomorphic as an additive group to C², which is a C-vector space. So R⁴ can also be considered a C-vector space. In this case, the definition is fairly straightforward. The same arguments would apply to R^2k, more generally.

R³ happens to be isomorphic to , say, R² as an additive group. This shows that R³ can in principle be endowed with a C-vector space structure. On the other hand, the isomorphisms that exist between R³ and R² cannot be defined easily, so there is no "obvious" C-vector space structure on R³. Somebody more knowledgeable than I am about set theory and logic could probably even make precise the statement that there is no "privileged" C-vector space structure on R³.

My point here is that when I say that K "needs" to be a subfield of L to make an L-vector space into a K-vector space, I have in mind not a precise mathematical theorem, but merely the fact that otherwise there is no obvious way to define a K-structure. In the particular case of F₂, however, the necessary and sufficient condition I stated was intended as a theorem of this nature.

It seems to me from a Google search that most books on the subject of finite-dimensional vector spaces over finite fields are to do with the computer science applications you mentioned. The theory of finite-dimensional vector spaces is very similar regardless of the field, except that the theory on inner products and orthogonality is really specific to R and C. (Some of that theory can be replaced with the theory of "duality" in more general situations.) Unless you find a reference specifically suited to your topic, and if you have time, my suggestion would be to get enough background on groups, rings, fields and vector spaces that you're comfortable with this kind of generalization. Also, understanding finite fields well is likely to require some knowledge of the theory of field extensions, although there may be elementary approaches I'm not aware of that avoid this. One possible book would be "Undergraduate Algebra" by Serge Lang, which has a chapter on finite fields.

What's the definition of '$\in$'?

This is a binary relation, And I know in set theory(like ZFC) A n-nary relation $R$ on X can be defined as this way: $R$ is a relation on $X$ if $R$ has the property such that $R\subset X^{n}$.
But I don't think '$\in$' can be defined by this way because the concept 'element', 'set', 'subset' in ZFC are all build up from this relation '$\in$'. And I can't find a statement given a definition to '$\in$' in first-order logic or other different formal systems. So help.TUOYUTSENG (talk) 15:55, 26 November 2011 (UTC)

The difficulty arises from the following convention. If R ⊆ S × T, we agree to write x R y, where x ∈ S and y ∈ T, as an abbreviation meaning that (x,y) ∈ R. You are correct that the notation x ∈ y already had a different meaning before this convention was introduced (even though this meaning is left undefined, since ∈ is a basic symbol of set theory). Writing x ∈ y does not mean we consider ∈ to be a set of pairs. It is simply that the notations are similar.

If necessary, we could define a set E ⊆ S × T as E = {(x,y) ∈ S × T : x ∈ y}. In that case, for x ∈ S and y ∈ T, the conventional notation x E y would be equivalent to x ∈ y. It would probably be unwise to use the symbol ∈ for the set E in this case. 96.46.192.31 (talk) 18:39, 26 November 2011 (UTC)

Thank you.TUOYUTSENG (talk) 05:37, 27 November 2011 (UTC)