Spans

**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:
\[\text{span}\{\mathbf{v}_1, \ldots, \mathbf{v}_k\} = \{c_1\mathbf{v}_1 + \cdots + c_k\mathbf{v}_k : c_i \in \mathbb{F}\}\]

The span is always a subspace of the containing vector space.

**Example:** In \(\mathbb{R}^3\), the span of \((1,0,0)\) and \((0,1,0)\) is the entire \(xy\)-plane:
\[\text{span}\{(1,0,0), (0,1,0)\} = \{(x, y, 0) : x, y \in \mathbb{R}\}\]

**Example:** A single nonzero vector \(\mathbf{v}\) spans a line through the origin: \(\text{span}\{\mathbf{v}\} = \{c\mathbf{v} : c \in \mathbb{R}\}\).

**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 \(V\).Copy