Span of a set of Vectors

<aside> 💡 The span of a set $S$ (of vectors) is defined as the set of all finite linear combinations of elements(vectors) of $S$, denoted by $span(S)$. i.e., $span(S)= \{ \sum_{i=1}^na_iv_i\in V |a_1,a_2,...,a_n\in \R \}$

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Example:

Let $S=\{ (1,0,0),(0,1,0)\} \subset \R^3$.Then:

$span(S)=\{a(1,0,0)+b(0,1,0) |a,b\in \R ~\}=\{(a,b,0)|a,b\in \R\}$

Spanning Set for a Vector Space

<aside> 💡 Let $V$ be a vector space. A set $S \subseteq V$ is a spanning set for $V$ if $span(S)=V.$

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Example: Adding vectors to obtain a spanning set for $\R^3$

What is a basis?

<aside> 💡 A basis $B$ of a vector space $V$ is a linearly independent subset of $V$ that spans $V$.

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