Linear combinations and spanning

Overview and basic definitions

Definition: Linear combination, span

Let $F$ be a field and $V$ be an $F$-vector space. A linear combination of a sequence of vectors $v_1,\dots ,v_d\in V$ is another vector of the form $t_1{v}_1+\cdots +t_d{v}_d,$ where each $t_i\in F$.

The span of the set $\{v_1,\dots ,v_d\}$ is the set of all linear combinations $\text{span}\{v_1,\dots ,v_d\}=\{t_1{v}_1+\cdots +t_d{v}_d:t_i\in f\text{ for all }i\},$ with $\text{span}\varnothing =\{0\}$.

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Properties of span

Proposition: Minimality of the spanning subspace

Let $v_1,\dots ,v_d\in V$ be a sequence of vectors.

• (i) $\text{span}\{v_1,\dots ,v_d\}$ is a vector subspace of $V$ containing $v_i$ for every $i$;
• (ii) If $W\subseteq V$ is another subspace that contains $v_1,\dots ,v_d$, then $\text{span}\{v_1,\dots ,v_d\}\subseteq W.$ That is, $\text{span}\{v_1,\dots ,v_d\}$ is the smallest vector subspace of $V$ containing the sequence $v_1,\dots ,v_d$;
• (iii) For every $v\in V$, we have $\text{span}\{v_1,\dots ,v_d\}\subseteq \text{span}\{v_1,\dots ,v_d,v\},$ with equality if and only if $v\in \text{span}\{v_1,\dots ,v_d\}$.

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Code snippets

v_1, \ldots, v_d

t_1 v_1 + \cdots + t_dv_d

\textup{span} \{ v_1, \ldots, v_d \}