# Abstract Nonsense

## 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.

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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 $F$-space $\mathscr{V}$ one has that $\mathscr{V}\cong F^{\dim \mathscr{V}}$. But, really even thinking about vector spaces as really just coordinate spaces of the form $F^n$ 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 $\mathbb{R}$). Probably the most formal, sterile way of thinking about vector spaces is just elements of $F$ with placeholders. Something like $\mathcal{P}_n$ (the set of all polynomials of degree less than or equal to $n$) 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 $n$ dimensional vector space. But, one doesn’t want to screw around with the notation of making it act on $F^n$, 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.

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

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Free Vector Space Definition

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Let $X$ be a finite set and $F$ a field. We define the set $F[X]$ to be the set of all formal sums of the form $\displaystyle \sum_{x\in X}f_x x$ where $f_x\in F$ for every $x\in X$. We then define the sum‘ of two of these formal sums as

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$\displaystyle \left(\sum_{x\in X}f_x x\right)+\left(\sum_{x\in X}g_x x\right)=\sum_{x\in X}(f_x+g_x)x$

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We define the scalar product of some $f\in F$ and a formal sum as

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$\displaystyle f\sum_{x\in X}f_x x=\sum_{x\in X}(f f_x)x$

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With this definition of addition and scalar multiplication of elements of $F[X]$ we call $F[X]$ the free vector space of $X$ over $F$. Suppose that we are given an indexing set $I$. We can then choose a formal object $x$ to ‘append’ the indices to, namely we can form the set $I_x=\left\{x_i:i\in I\right\}$ and we can then define the free vector space of the indexing set $I$ over $F$ to be $F[I_x]$. Evidently then $F[X]$ is a $F$-space with $\dim_F F[X]=\#(X)$.

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Why Anyone Cares

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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 $I$ and a function $f:I\to I$. We can then form the free vector space $F[I_x]$ and define the function $T_f:F[I_x]\to F[I_x]$ by

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$\displaystyle T_f\left(\sum_{i\in I}z_i x_i\right)=\sum_{i\in I}z_i x_{f(i)}$.

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$T_f$ is easily seen to be a linear transformation.

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For example one could (and is often–almost always) define the group algebra $\mathcal{A}(G)$ to be $\mathbb{C}[G]$. Then, the left regular representation $L$ can be thought of as $L(h)=T_{f_h}$ where $f_h:G\to G:g\mapsto gh$.