Span (linear algebra)
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Contents
Definition
Let [ilmath](V,\mathbb{F})[/ilmath] be a vector space over a field [ilmath]\mathbb{F} [/ilmath] and let [ilmath]\{v_\alpha\}_{\alpha\in I}\subseteq V[/ilmath] be an arbitrary collection of vectors of [ilmath]V[/ilmath]. The span of [ilmath]\{v_\alpha\}_{\alpha\in I} [/ilmath], denoted:
- [ilmath]\text{Span}(\{v_\alpha\}_{\alpha\in I})[/ilmath] for an arbitrary collection, or
- [ilmath]\text{Span}(v_1,\ldots,v_k)[/ilmath] for a finite collection
is defined as follows:
- [math]\text{Span}(\{v_\alpha\}_{\alpha\in I}):\eq\Bigg\{\underbrace{\sum_{\alpha\in I}\lambda_\alpha v_\alpha}_{\text{Linear combination} }\ \Bigg\vert\ \{\lambda_\alpha\}_{\alpha\in I}\in\underbrace{\big\{\{\lambda_\alpha\}_{\alpha\in I}\in\mathbb{F}^I\ \big\vert\ \overbrace{\vert\{\lambda_\alpha\ \vert\ \alpha\in I\wedge\lambda_\alpha\neq 0\}\vert\in\mathbb{N} }^{\text{There are only finitely many non-zero terms} }\big\} }_{\text{The set of }I\text{-indexed scalars such that}\{\lambda_\alpha\}_{\alpha\in I}\text{ only has finitely many non-zero terms} } \Bigg\} [/math][Note 1]
For a finite collection, [ilmath]\{v_1,\ldots,v_k\} [/ilmath] this simplifies to:
- [ilmath]\text{Span}(v_1,\ldots,v_n):\eq\left\{\left.\sum_{i\eq 1}^k\lambda_i v_i\ \right\vert\ (\lambda_i)_{i\eq 1}^k\in\mathbb{F}\right\} [/ilmath]
See linear combination for details of why we need the "finitely many part"
Caveats
Caveat:There are a few problems here
- Ordered basis - in the set [ilmath]\{v_1,\ldots,v_n\} [/ilmath] there is no order, we really mean [ilmath](v_i)_{i\eq 1}^n[/ilmath] - this implies order. Also for an arbitrary collection, tuple notation doesn't make sense unless there's an ordering in play. This needs to be "united"
Notes
- ↑ Remember that the "vector addition" is a binary function on [ilmath]V[/ilmath]. It's also associative so [ilmath](u+v)+w\eq u+(v+w)[/ilmath] which makes things easier. However we can only do this finitely many times ultimately. So we use the following abuse of notation:
- We may define arbitrary sums on the condition that it only has finitely many non zero terms. We use the fact that zero vector is the additive identity (and thus [ilmath]0+v\eq v[/ilmath]) to "pretend" we included them, they have no effect on the summation's value.
- Notations like [ilmath]\sum^\infty_{n\eq 0}v_n[/ilmath] make no sense in a vector space. This definition of limit requires a topological space. That is where the "tending towards" comes from. [ilmath]\mathbb{R} [/ilmath] is a vector space (as is [ilmath]\mathbb{R}^n[/ilmath] and so forth) but also a metric space (infact an inner product space) and thus a topological space.
