Span, linear independence, linear dependence, basis and dimension

This article includes information on linear dependence and independence an introduction and discussion of these very important concepts can be found on the Basis and coordinates page

Span

Definition

Given a set of vectors $S$ in a vector space $(V,F)$

$$\text{Span}(S)=\left\{\sum^n_{i=1}\lambda v_i|n\in\mathbb{N},\ v_i\in S,\ \lambda_i\in F\right\}$$

It is very important that only finite linear combinations are in the span.

Linear Dependence

A set $E$ in a vector space $(V,F)$ is linearly dependent if for any finite collection of elements of $E$ that finite collection is linearly dependent

That is,

$$\forall n\in\mathbb{N}$$ given a subset $$\{e_1,...,e_n\}\subset E$$

There are solutions to

$$\sum^n_{i=1}e_i\alpha_i=0$$ where the $$\alpha_i\in F$$ are not all zero.

Linear Independence

If a set is not linearly dependent it is linearly independent, but formally:

For all finite subsets of a set $E$, we have only

$$\alpha_i=0\forall i$$ as solutions to $$\sum^n_{i=1}e_i\alpha_i=0$$

Basis

Usually a basis will be a finite set, for example,

$$\{(1,0),(0,1)\}$$ is a basis of $\mathbb{R}^2$.

Finite case

Given a finite set $B\subset V$, $B$ is a basis of $V$ if

$$\text{span}(B)=V$$ and $B$ is linearly independent.

Infinite case

A Hamel basis is any linearly independent subset of $V$ that spans $V$ - where linearly independent is given as above.

The definition of independence varies slightly from how it is usually given (I explicitly say for all finite subsets) it is just a stronger form.

Example

The set

$$E=\{1,x,x^2,x^3,...,x^i,...\}$$ is a Hamel basis for the space of all polynomials

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