Vector Spaces are Free Objects

by Musa Al-hassy   Last Updated September 21, 2018 20:20 PM

Warning: I know little linear algebra and my assertions below may all be incorrect. I am interested in lists --i.e., free monoids-- and my interest has led me to [finite-dimensional] vector spaces.

Let 𝓀 be any field, then we can freely equip any [finite] set S with a 𝓀-vector space $$F(S) = (S → 𝓀)$$, a set of functions where vector addition is performed pointwise, and likewise for the other vector operations. Moreover any function $$S → |𝒱|$$ from $$S$$ to the underlying set of a vector space 𝒱 can be extended to a linear operator $$F(S) → 𝒱$$ that behaves the same on elements of $$S$$ construed as vectors of $$F(S)$$.

That is, $$F$$ is a “free functor”: It constructs the least vector space that contains a (copy of a) given set.

It is known that every vector space 𝓥 admits a basis β and so is isomorphic to $$F(β)$$ --which is essentially the dual space of 𝓥. That is, every vector space is free; i.e., is in the image of functor $$F$$.

Of-course we have to choose a basis to realise a vector space as a free object; there is no canonical basis.

( Aside: With this in-hand, we can easily prove equi-dimensional spaces are necessarily isomorphic: That they're equi-dimensional means their basis are in bijection, but free objects are unique up to unique isomorphism and so their F-images must be unique up to unique isomorphism as well. That is, $$𝓥 ≅ 𝓀^{dim 𝒱} = 𝓀^{dim 𝒱} ≅ 𝓦$$. Neat stuff! )

Questions:

1. Intuitively why is it that vector spaces admit basis.

• I'm not interested in a proof.
• For example, why is it that monoids or rings are not always generated by some set, but vector spaces are. The definition of a vector space does not immediately give rise to a basis.
2. All [finite dimensional] vector spaces admit basis and so the objects of 𝑽𝑬𝑪, the category of finite dimensional vector spaces and linear operators, are all images of a free functor. (Assuming we have a way to choose basis.)

• Is there a name for categories whose objects are all isomorphic to an image of a free functor?
• What other examples of such categories are there?
• Are there any constructions that produce such categories.
3. Lists over a set $$S$$ are isomorphic to $$⋃_{n ∈ ℕ} Sⁿ$$ and provide the free monoid over a given set $$S$$. Why is it that lists, whence monoids, are far more prevalent than their vector space counterparts within computing science.

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