Catalog of Definitions from Linear Algebra.

Definition 1. Vector Space
§1.2
A vector space (or linear space) Vover a field F consists of a set on which twooperations (called addition andscalar multiplication) are defined so thatfor each pair of elements x,y∈Vthere is a unique element x+y∈V, andfor each a∈F and x∈Vthere is a unique element ax∈V,such that the following conditions hold:(VS 1)∀x,y∈V, x+y=y+x(commutativity of addition)(VS 2)∀x,y,z∈V, (x+y)+z=x+(y+z)(associativity of addition)(VS 3)∃ 0∈V s.t. x+0=x for each x∈V(VS 4)∀x∈V, ∃ y∈V s.t. x+y=0(VS 5)∀x∈V, 1x=x(VS 6)∀a,b∈F, ∀x∈V, (ab)x=a(bx)(VS 7)∀a∈F, ∀x,y∈V, a(x+y)=ax+ay(VS 8)∀a,b∈F, ∀x∈V, (a+b)x=ax+bxThe elements x+y and ax are calledthe sum of x and y and the product of a and x.The elements of F are called scalars,and the elements of V are called vectors.\text{A } \textbf{vector space} \text{ (or } \textbf{linear space}\text{) } V \\ \text{over a field } \mathbb{F} \text{ consists of a set on which two} \\ \text{operations (called } \textbf{addition} \text{ and} \\ \textbf{scalar multiplication}\text{) are defined so that} \\ \text{for each pair of elements } x, y \in V \\ \text{there is a unique element } x + y \in V\text{, and} \\ \text{for each } a \in \mathbb{F} \text{ and } x \in V \\ \text{there is a unique element } ax \in V\text{,} \\ \text{such that the following conditions hold:} \\[1em] \begin{array}{ll} \htmlId{vs-1}{\text{(VS 1)}} & \forall x, y \in V,\ x + y = y + x \\ & \text{(commutativity of addition)} \\[0.4em] \htmlId{vs-2}{\text{(VS 2)}} & \forall x, y, z \in V,\ (x + y) + z = x + (y + z) \\ & \text{(associativity of addition)} \\[0.4em] \htmlId{vs-3}{\text{(VS 3)}} & \exists\, 0 \in V \text{ s.t. } x + 0 = x \text{ for each } x \in V \\[0.4em] \htmlId{vs-4}{\text{(VS 4)}} & \forall x \in V,\ \exists\, y \in V \text{ s.t. } x + y = 0 \\[0.4em] \htmlId{vs-5}{\text{(VS 5)}} & \forall x \in V,\ 1x = x \\[0.4em] \htmlId{vs-6}{\text{(VS 6)}} & \forall a, b \in \mathbb{F},\ \forall x \in V,\ (ab)x = a(bx) \\[0.4em] \htmlId{vs-7}{\text{(VS 7)}} & \forall a \in \mathbb{F},\ \forall x, y \in V,\ a(x + y) = ax + ay \\[0.4em] \htmlId{vs-8}{\text{(VS 8)}} & \forall a, b \in \mathbb{F},\ \forall x \in V,\ (a + b)x = ax + bx \end{array} \\[1em] \text{The elements } x + y \text{ and } ax \text{ are called} \\ \text{the } \textbf{sum} \text{ of } x \text{ and } y \text{ and the } \textbf{product} \text{ of } a \text{ and } x\text{.} \\ \text{The elements of } \mathbb{F} \text{ are called } \textbf{scalars}\text{,} \\ \text{and the elements of } V \text{ are called } \textbf{vectors}\text{.}
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}