> [!NOTE] Definition (Span of subset of vector space) > Let $S \subset V$ be a non-empty subset of a [[Vector spaces|vector space]] $V$. The span of $S$, denoted $\langle S\rangle$ is $\langle S\rangle = \{ \lambda_{1}v_{1} +\dots + \lambda_{s} v_{s} \mid v_{1},\dots, v_{s}\in S \text{ and } \lambda_{1},\dots,\lambda_{s} \in \mathbb{R} \}$ > We define the span of the empty subset $S=\emptyset \subset V$ to be $\langle \emptyset \rangle = \{ 0_{V} \}$. > > [!Definition] Definition (Linear Combination) > For any [[Vectors|vectors]] $\underline{v_{1}}, \underline{v_{2}}, \dots, \underline{v_{s}} \in \mathbb{F}^{n}$ and scalars $\lambda_{1}, \lambda_{2},\dots,\lambda_{s} \in \mathbb{F}^{n}$, the expression $\sum_{i=1}^{s} \lambda_{i} \underline{v}_{i} \in \mathbb{F}^{n}$is called a linear combination of $\underline{v_{1}}, \underline{v_{2}},\dots,\underline{v_{s}} \in \mathbb{F}^{n}$ > [!info] Definition (Spanning set of FDVS) > Suppose $V$ is [[Vector spaces|finite dimensional]]. A finite subset $S\subset V$ is a spanning set of $V$ iff $\langle S \rangle= V$. ^5c027b > [!Example] > The [[Standard basis of real n-space]] is spanning set of $\mathbb{R}^n$. # Properties > [!info] Bais > If a spanning set is linearly independent, then it is called a [[Basis of Vector Space|basis]]. > [!NOTE] Theorem (Span of Linearly independent vectors) > See [[Linear Independence#^c97b84|span of linearly independent vectors]]. # Applications - [[Column Span of Real Matrix]].