Difference between revisions of "Formal linear combination/Definition"

Latest revision as of 15:36, 24 December 2016

Definition

Let $S$ be a set and let $\mathbb{F}$ be a field^{[Note 1]}, then^[1]:

• Informally^{[Note 2]} 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^[1]:
  • $f:S\rightarrow\mathbb{F}$ such that $\big\vert\{s\in S\ \vert\ f(s)\neq 0\}\big\vert\in\mathbb{N}$^{Warning:[Note 3], [Note 4]} (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.

Notes

1. Jump up ↑ We could probably step back and define this the same way on a ring, as a field is itself a ring it'd be the same thing. Modules are very similar to vec spaces after all
2. Jump up ↑ Ignore the "informally a formal ..."
3. Jump up ↑ Caveat:Be aware that $\big\vert\{f(s)\neq 0\ \vert\ s\in S\}\big\vert\in\mathbb{N}$ is different to $\big\vert\{s\in S\ \vert\ f(s)\neq 0\}\big\vert\in\mathbb{N}$ as the first set is the number of non-zero things the function maps to not the number of things that map to non-zero things. For example:
   • if we take the function $f:\mathbb{N}\rightarrow\mathbb{N}$ given by $f:n\mapsto\left\{\begin{array}{lr}0 & \text{if }n\text{ is odd}\\ 1 & \text{otherwise}\end{array}\right.$ then $\big\vert\{f(n)\neq 0\ \vert\ n\in \mathbb{N}\}\big\vert\in\mathbb{N}$ indeed holds, as $\vert\{1\}\vert\eq 1$ however $\big\vert\{n\in \mathbb{N}\ \vert\ f(n)\neq 0\}\big\vert\in\mathbb{N}$ doesn't hold as the set of even numbers is not finite.
4. Jump up ↑ Zero here denotes the "additive identity" of the field, $\mathbb{F}$

References

1. ↑ ^{Jump up to: 1.0 1.1} Introduction to Smooth Manifolds - John M. Lee