## Free Vector Spaces

**Point of post: **In this post we discuss the notions of free vector spaces. In particular we discuss their construction and where they naturally come up in the ‘real world’ of mathematics.

*Motivation*

In essence linear algebra (in the sense of finite dimensional vector spaces) is a very simple subject. Namely, it’s trivial that given any finite dimensional -space one has that . But, really even thinking about vector spaces as really just coordinate spaces of the form doesn’t catch the full feel of what we’re really studying since this adds a geometric aspect to the mix (in the case of ). Probably the most formal, sterile way of thinking about vector spaces is just elements of with placeholders. Something like (the set of all polynomials of degree less than or equal to ) is an excellent example of the idea–except we might get caught up in thinking of the polynomials as functions instead of just placeholders. This is where free vector spaces come into play. Namely, we’ll do precisely what we said before–make a vector space out of formal symbols. This comes up more often than one may think. Namely, often one has a ‘linear transformation’ which should act on an dimensional vector space. But, one doesn’t want to screw around with the notation of making it act on , etc–so one just defines a free vector space. Most often this comes up when what really happens is that one has a set of indices and a function on the indices and one define the free vector space over the indices and then bam, suddenly your function on the indices naturally becomes a linear transformation.

It is interesting to compare this intuitive idea to the formal definitions really going on in the background, namely the idea of free modules.

*Free Vector Space Definition*

Let be a finite set and a field. We define the set to be the set of all formal sums of the form where for every . We then define the *sum*‘ of two of these formal sums as

We define the *scalar product *of some and a formal sum as

With this definition of addition and scalar multiplication of elements of we call the *free vector space of over .* Suppose that we are given an indexing set . We can then choose a formal object to ‘append’ the indices to, namely we can form the set and we can then define the *free vector space of the indexing set over *to be . Evidently then is a -space with .

*Why Anyone Cares*

The above may, at first, seem quite pedantic and useless. But, one encounters free vector spaces over sets quite often (for example in the formal definition of the tensor prodcuct). But, a particularly common place they come up is the following situation. Suppose that one is given an indexing set and a function . We can then form the free vector space and define the function by

.

is easily seen to be a linear transformation.

For example one could (and is often–almost always) define the group algebra to be . Then, the left regular representation can be thought of as where .

[…] 6: Let be a finite group and . Let be the distinct cosets of . Let be thefree complex vector space over the symbols . Then, for define on by if . The map is a representation of (you do not […]

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[…] we are adding them in the sense of taking the free vector space (which of course we can identify with the group algebra by ) and is the sign function. We […]

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