Is the Field of Reals in and of itself a Vector Space?

[Rising Eagle]

There is a difference between the multiplication op in the Field of Reals and the Scaling op in the Set of Vectors. In the Field, the op is a closed mapping between two members within the same Set, whereas in the Set of Vectors, the op is a mapping between a member of the Set of Vectors and a member from the Set of Scalars (a different set entirely) that is closed within the Set of Vectors. The question arises: if we now let the Set of Vectors become one and the same with the Set of Scalars (i.e., the Field of Reals), do we indeed still have a Vector Space that is also a Field or do we just have a Field? If we just have a Field, is it still a full fledged Vector Space? I am uncertain of the answer as all Vector expressions would resolve to a single numerical value, collapsing the internal mechanical workings that make a Vector Space useful as a model for Physical processes.

Additional question: If the Field of Reals is duplicated so that we have a pair of twin Fields and we can, in theory, keep them entirely separate, can we answer the above question by saying "we have a Field of Reals as a Vector Space over a twin Field of Reals and we genuinely have a Vector Space in its conventional form"?

 

[Number Nine]

If we want to be pedantic, then no, in the sense that a field is not the same as a vector space. It can however, be considered a vector space over itself.
Useful note: Given a field F and a subfield E, then F is a vector space over E.

 

[jgens]

The usual multiplication on the field of reals is the same as the usual scalar multiplication when the reals are considered as a vector space over themselves. If you are unconvinced of this fact, then write the map out in terms of set level data. The only distinction you can make is that technically a field is a quintuple $(F,+,0,\cdot,1)$ whereas a vector space is an octuple $(V,+_V,0_V,F,+_F,0_F,\cdot_F,1_F)$. However, it really is not useful to view fields and vector spaces as tuples, hence the identification of the field of reals as the vector space of reals over themselves.

Note: I am aware of other formalisms for fields and vector spaces, but the one presented here is sufficient to get the point across.

 

[SteveL27]

Additional question: If the Field of Reals is duplicated so that we have a pair of twin Fields and we can, in theory, keep them entirely separate, can we answer the above question by saying "we have a Field of Reals as a Vector Space over a twin Field of Reals and we genuinely have a Vector Space in its conventional form"?

Nothing says that the field of scalars and the set of vectors can't be the same underlying set. In this case they are. The field is the set of reals with the usual multiplication and addition; and the set of vectors is the set of reals with the usual addition.

If it helps to think about it this way, the set of reals with the usual addition and multiplication is a field; and the same set of reals with just the usual addition is an additive Abelian group. So it's not really a "twin" set, it's the same set being used twice.

 

[blue_raver22]

The Field of Reals, by definition, is a set of numbers that follows certain properties and operations, such as addition and multiplication. It is not inherently a vector space, as a vector space requires additional structure and operations.

In a vector space, the multiplication operation is defined as scalar multiplication, where a scalar (a member of the field of scalars) is multiplied by a vector (a member of the vector space), resulting in a new vector. This operation is not the same as the multiplication operation in the Field of Reals, as it involves two different sets (the set of scalars and the set of vectors) and a different type of mapping.

If we were to merge the Field of Reals and the set of vectors, as suggested in the question, we would not have a vector space anymore. We would simply have a field, as all vector expressions would collapse to a single numerical value. This would not be a full-fledged vector space, as it would lack the additional properties and operations that define a vector space.

As for the additional question, if we were to have a pair of twin Fields of Reals that are kept entirely separate, we would still not have a vector space. Each field would still function as a field, with its own set of operations and properties. We cannot create a vector space by simply duplicating a field, as a vector space requires specific structure and operations that are not present in a field alone.