Catalog of Definitions from Linear Algebra.

Definition 2. Subspace
§1.3
W⊆V is a subspace of V if W is a vector space over Fwith the operations of addition and scalarmultiplication defined on V.Equivalently, W is a subspace of V iff the following hold:(a)(Closed under addition)x+y∈W whenever x∈W and y∈W(b)(Closed under scalar multiplication)cx∈W whenever c∈F and x∈W(c)0∈W(d)Each vector in W has an additive inverse in WW \subseteq V \text{ is a } \textbf{subspace} \text{ of } V \text{ if } W \text{ is a vector space over } \mathbb{F} \\ \text{with the operations of addition and scalar} \\ \text{multiplication defined on } V. \\[0.5em] \text{Equivalently, } W \text{ is a subspace of } V \text{ iff the following hold:} \\[1em] \begin{array}{ll} \text{(a)} & \text{(Closed under addition)} \\ & x + y \in W \text{ whenever } x \in W \text{ and } y \in W \\[0.6em] \text{(b)} & \text{(Closed under scalar multiplication)} \\ & cx \in W \text{ whenever } c \in \mathbb{F} \text{ and } x \in W \\[0.6em] \text{(c)} & 0 \in W \\[0.6em] \text{(d)} & \text{Each vector in } W \text{ has an additive inverse in } W \end{array}