# Spans

Put content here**Definition:** The *span* of a set of vectors \(\{\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_k\}\) is the set of all possible linear combinations of those vectors:
\[\text{span}\{\mathbf{v}_1, \ldots, \mathbf{v}_k\} = \{c_1\mathbf{v}_1 + \cdots + c_k\mathbf{v}_k : c_i \in \mathbb{F}\}\]
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The span is always a subspace of the containing vector space.
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**Example:** In \(\mathbb{R}^3\), the span of \((1,0,0)\) and \((0,1,0)\) is the entire \(xy\)-plane:
\[\text{span}\{(1,0,0), (0,1,0)\} = \{(x, y, 0) : x, y \in \mathbb{R}\}\]
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**Example:** A single nonzero vector \(\mathbf{v}\) spans a line through the origin: \(\text{span}\{\mathbf{v}\} = \{c\mathbf{v} : c \in \mathbb{R}\}\).
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**Key property:** If \(\text{span}\{\mathbf{v}_1, \ldots, \mathbf{v}_k\} = V\), then \(\{\mathbf{v}_1, \ldots, \mathbf{v}_k\}\) is called a *spanning set* for \(V\).

# Parents

* Coordinate vector spaces