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).