Free vector space generated by

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Definition

Let $S$ be a set and let $\mathbb{F}$ be a field. The "free vector space over $\mathbb{F}$ generated by $S$", denoted $\mathcal{F}_\mathbb{F}(S)$ or $\mathcal{F}(S;\mathbb{F})$ is defined as follows:$\newcommand{\freevec}[2][\mathbb{F}]{\mathcal{F}(#2;#1)}$

• $\freevec{S}$ is the set of all formal linear combinations of elements of $S$
• $\freevec{S}$ is a vector space under the operations of pointwise addition and the obvious scalar multiplication as follows:
  1. Addition: let $f,g\in\freevec{S}$ then $(f+g):S\rightarrow\mathbb{F}$ is given by $(f+g):s\mapsto f(s)+g(s)$
     • This addition is defined as both $f(s),g(s)\in\mathbb{F}$ and a field has addition
  2. Scalar multiplication: let $\lambda\in\mathbb{F}$ and $f\in\freevec{S}$ be given, then we define $(\lambda f):S\rightarrow\mathbb{F}$ by $(\lambda f):s\mapsto \lambda f(s)$
     • Again, $\lambda,f(s)\in\mathbb{F}$ and as $\mathbb{F}$ is a field, this notion of multiplication is defined.

Formal linear combinations

Let $S$ be a set and let $\mathbb{F}$ be a field, then:

• Informally a formal linear combination is an expression of the form:
  • $$\lambda_1 s_1 + \lambda_2 s_2 + \cdots + \lambda_{m-1} s_{m-1} + \lambda_m s _m\eq \sum^m_{i\eq 1}\lambda_i s_i$$
    • for some $m\in\mathbb{N}$, some $\lambda_i\in\mathbb{F}$ and some $s_i\in S$
  • We never actually define $\lambda s$ (the multiplication of $s\in S$ by a $\lambda\in\mathbb{F}$) nor do we define any sort of "addition" operation, this is simply an expression.
  • We want it to behave as a linear combination normally would, i.e.:
    1. For example: $$\big(\lambda_1 s_1+\lambda_2 s_2\big)+\big(\mu_1 s_1 + \mu_2 s_3 + \mu_3 s_4\big)\eq \alpha s_1 + \lambda_2 s_2 + \mu_2 s_3 + \mu_3 s_4$$ say, where $\alpha:\eq \lambda_1 + \mu_1$ - which is defined as $\lambda_i,\mu_j\in\mathbb{F}$ remember. And
    2. For example: $$\mu(\lambda_1 s_1 + \cdots + \lambda_n s_n)\eq \alpha_1 s_ 1 + \cdots + \alpha_n s_n$$ where $\alpha_i:\eq \mu\lambda_i$ - which is defined as $\lambda_i,\mu\in\mathbb{F}$ of course.
  • Even though we can never give it a value
• Formally, a formal linear combination of elements of $S$ with respect to the field $\mathbb{F}$ is a function:
  • $f:S\rightarrow\mathbb{F}$ such that $\big\vert\{s\in S\ \vert\ f(s)\neq 0\}\big\vert\in\mathbb{N}$^Warning:, (where $\big\vert\cdot\big\vert$ denotes cardinality)
    • That is to say $f$ takes non-zero values a finite number of times only, it is zero "almost everywhere"
  • $f$ represents $$\sum_{s\in S}f(s)s$$ as a linear combination, even if the sum were formally defined to have meaning, we still use the usual abuse of notation when only finitely many elements of the summation are non-zero whereby $$\sum_{s\in S}f(s)s$$ means $$\sum_{\begin{array}{cc}s\in S\\ f(s)\neq 0\end{array} }f(s)s$$ , hence the requirement that $f$ only maps finitely many things to non-zero things.

Characteristic property

Characteristic property of the free vector space/Statement