span

The set of every linear combination of a given collection of vectors.

The span of vectors $\mathbf v_1,\dots,\mathbf v_k$ in a vector space $V$ , written $\operatorname{span}(\mathbf v_1,\dots,\mathbf v_k)$ , is the set of every linear combination that can be formed from them: $\{c_1\mathbf v_1+\cdots+c_k\mathbf v_k : c_1,\dots,c_k\in\mathbb R\}$ . Span is always a subspace of $V$ , no separate check required — it contains $\mathbf 0$ and is closed under addition and scalar multiplication directly from the definition of a linear combination.