Definition:Vector Space/Definition 2
Definition
Let $\struct {K, +_K, \times_K}$ be a field whose unity is $1_K$.
Let $\struct {G, +_G}$ be an abelian group.
Let $\struct {\map {\mathrm {End} } G, +, \circ}$ be the endomorphism ring of $\struct {G, +_G}$ such that $I_G$ is the identity mapping.
Let $\cdot: \struct {K, +_K, \times_K} \to \struct {\map {\mathrm {End} } G, +, \circ}$ be a ring homomorphism from $K$ to $\map {\mathrm {End} } G$ which maps $1_K$ to $I_G$.
Then $\struct {G, +_G, \cdot, K}$ is a vector space over $K$ or a $K$-vector space.
Vector Space Axioms
The vector space axioms consist of the abelian group axioms:
\((\text V 0)\) | $:$ | Closure Axiom | \(\ds \forall \mathbf x, \mathbf y \in G:\) | \(\ds \mathbf x +_G \mathbf y \in G \) | |||||
\((\text V 1)\) | $:$ | Commutativity Axiom | \(\ds \forall \mathbf x, \mathbf y \in G:\) | \(\ds \mathbf x +_G \mathbf y = \mathbf y +_G \mathbf x \) | |||||
\((\text V 2)\) | $:$ | Associativity Axiom | \(\ds \forall \mathbf x, \mathbf y, \mathbf z \in G:\) | \(\ds \paren {\mathbf x +_G \mathbf y} +_G \mathbf z = \mathbf x +_G \paren {\mathbf y +_G \mathbf z} \) | |||||
\((\text V 3)\) | $:$ | Identity Axiom | \(\ds \exists \mathbf 0 \in G: \forall \mathbf x \in G:\) | \(\ds \mathbf 0 +_G \mathbf x = \mathbf x = \mathbf x +_G \mathbf 0 \) | |||||
\((\text V 4)\) | $:$ | Inverse Axiom | \(\ds \forall \mathbf x \in G: \exists \paren {-\mathbf x} \in G:\) | \(\ds \mathbf x +_G \paren {-\mathbf x} = \mathbf 0 \) |
together with the properties of a unitary module:
\((\text V 5)\) | $:$ | Distributivity over Scalar Addition | \(\ds \forall \lambda, \mu \in K: \forall \mathbf x \in G:\) | \(\ds \paren {\lambda + \mu} \circ \mathbf x = \lambda \circ \mathbf x +_G \mu \circ \mathbf x \) | |||||
\((\text V 6)\) | $:$ | Distributivity over Vector Addition | \(\ds \forall \lambda \in K: \forall \mathbf x, \mathbf y \in G:\) | \(\ds \lambda \circ \paren {\mathbf x +_G \mathbf y} = \lambda \circ \mathbf x +_G \lambda \circ \mathbf y \) | |||||
\((\text V 7)\) | $:$ | Associativity with Scalar Multiplication | \(\ds \forall \lambda, \mu \in K: \forall \mathbf x \in G:\) | \(\ds \lambda \circ \paren {\mu \circ \mathbf x} = \paren {\lambda \cdot \mu} \circ \mathbf x \) | |||||
\((\text V 8)\) | $:$ | Identity for Scalar Multiplication | \(\ds \forall \mathbf x \in G:\) | \(\ds 1_K \circ \mathbf x = \mathbf x \) |
Vector
Let $V$ be a vector space.
Any element $v$ of $V$ is called a vector.
Zero Vector
The identity of $\struct {G, +_G}$ is usually denoted $\bszero$, or some variant of this, and called the zero vector:
- $\forall \mathbf a \in \struct {G, +_G, \circ}_R: \bszero +_G \mathbf a = \mathbf a = \mathbf a +_G \bszero$
Note that on occasion it is advantageous to denote the zero vector differently, for example by $e$, or $\bszero_V$ or $\bszero_G$, in order to highlight the fact that the zero vector is not the same object as the zero scalar.
Also known as
A vector space is also sometimes called a linear space, especially when discussing the real vector space $\R^n$.
Some go further and refer to a linear vector space
The notation $\struct {G, +_G, \circ, K}$ can also be seen for this concept.
Also see
Sources
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Algebraic Concepts