Vector Space Algebra
Algebraically speaking, a vector space is defined over an underlying scalar field. A field offers the operations and properties of basic arithmetics; common examples are rational, real or complex numbers. The definition of field is in turn based on the definition of group:
Algebraic Groups
A group G = (G, *, e) consists of a set of elements G, an inner operator *, and an identity element e, with the following properties:
- For all a, b in G: a * b is in G; i.e., * is an inner operator on G.
- For all a, b, c in G: a * (b * c) = (a * b) * c; i.e., operator * is associative.
- e is an element of G.
- For all a in G: e * a = a; i.e., e is left-identity element of operator *.
- For all a in G: There exists a-1 in G such that a-1 * a = e; i.e., each element of G has a left-inverse element under operator *.
If, additionally,
- for all a, b in G: a * b = b * a; i.e., operator * is commutative,
then G is called a commutative (or abelian) group.
The group axioms only mention left-identity and left-inverse elements. It can be derived from the axioms that the left-inverse a-1 of an element a is also its right-inverse: a * a-1 = e * a * a-1 = ((a-1)-1 * a-1) * a * a-1 = (a-1)-1 * (a-1 * a) * a-1 = (a-1)-1 * e * a-1 = (a-1)-1 * a-1 = e; and that the left-identity element e is also the right-identity element: a * e = a * (a-1 * a) = (a * a-1) * a = e * a = a. Thus, we speak only of identity element e and inverse element a-1.
Note: In a more functional notation, a group can also be written as G = (G, *, e, -1), where G is still a set of elements and * is an inner operator, i.e., a function *: G x G -> G written in infix notation, but now e is a (constant) function e: G whose value is the identity element of operator *, and -1 is a function -1: G -> G written in postfix notation, such that the value of a-1 is the inverse element of a under operator *.
Examples of groups are (R, +, 0), the group of real numbers with addition, and (R\{0}, *, 1), the group of real numbers (without zero) with multiplication. Note that (R, *, 1) is not a group since zero does not have an inverse element.
Algebraic Rings
Sets of elements become more interesting if there is more than one operation defined on them. A ring R = (R, +, *, 0) consists of a set of elements R, two inner operators + and *, and an identity element 0, with the following properties:
- (R, +, 0) is a commutative group as defined above, with inner operator + and identity element 0. The inverse of an element a in R under operator + is denoted as -a.
- For all a, b in R: a * b is in R; i.e., * is an inner operator on R.
- For all a, b, c in R: a * (b * c) = (a * b) * c; i.e., operator * is associative.
- For all a, b, c in R: (a + b) * c = (a * c) + (b * c); i.e., operators + and * are left-distributive.
- For all a, b, c in R: a * (b + c) = (a * b) + (a * c); i.e., operators + and * are right-distributive.
If, additionally,
- for all a, b in R: a * b = b * a; i.e., operator * is commutative,
then R is called a commutative ring.
It follows from the ring axioms that for all a in R: a * 0 = 0 * a = 0. Some rings exhibit the property that for all a, b in R: a * b = 0 => a = 0 or b = 0, or, in other words, the product of two non-zero elements is always non-zero. Such rings are called integral domains.
An examples of a ring is (R, +, *, 0), the ring of real numbers with usual addition and multiplication. This ring is also an integral domain.
Algebraic Rings With Unity
A special case of ring occurs if the second ring operator, has an identity element as well. A ring with unity R = (R, +, *, 0, 1) consists of a set of elements R, two inner operators + and *, and two identity elements 0 and 1, with the following properties:
- (R, +, *, 0) is a ring as defined above.
- For all a in R: a * 1 = 1 * a = a; i.e., 1 is identity element of operator *.
Incidentally, if R contains at least one element that is not equal to 0, then 1 and 0 must be different elements.
The ring (R, +, *, 0, 1) is a ring with unity.
Algebraic Fields
Though rings with unity may contain elements that have inverses under operator *, this is not required by the ring axioms. A field F = (F, +, *, 0, 1) consists of a set of elements F, two inner operators + and *, and two identity elements 0 and 1, with the following properties:
- (F, +, *, 0, 1) is a commutative ring with unity.
- 0 /= 1; i.e., F contains at least one non-zero element.
- For all a in F\{0}: There exists b in F such that a * b = b * a = 1; i.e., each element of F except zero has an inverse element under operator *. The inverse of an element a in F\{0} under operator * is denoted as a-1.
From the above axioms follows that F is an integral domain: Let a, b in F be any non-zero elements, and assume that a * b = 0. Since a /= 0, it has an inverse a-1. Multiplying both sides of a * b = 0 with a-1 yields a-1 * a * b = a-1 * 0, which is equivalent to b = 0, which violates the assumption that b /= 0. This also implies that for any field F = (F, +, *, 0, 1) the construct (F\{0}, *, 1) is a group.
Examples of fields are rational, real and complex numbers; each with the usual addition and multiplication operators.
Algebraic Vector Spaces
If F = (F, +, *, 0, 1) is a field, then a vector space over F is a construct V = (V, +, *, 0) that consists of a set of elements V, an inner operator +, an outer operator * and an identity element 0, with the following properties:
- (V, +, 0) is a commutative group. The elements of V are called vectors, 0 is called the zero or null vector, and the inverse of a vector v under operator + is denoted by -v.
- For all a in F, v in V: a * b is in V; i.e., * is an outer operator on F and V. Operator * is called scalar multiplication.
- For all a, b in F, v in V: (a * b) * v = a * (b * v); i.e., the two operators * (one in F, the other in V) are associative.
- For all v in V: 1 * v = v; i.e., the multiplicative identity element of F is also identity element with respect to operator * in V.
- For all a, b in F, v in V: (a + b) * v = (a * v) + (b * v); i.e., operators + and * are left-distributive.
- For all a in F, v, w in V: a * (v + w) = (a * v) + (a * w); i.e., operators + and * are right-distributive.
The above axioms imply the following additional relationships between a vector space and its underlying scalar field:
- For all v in V: 0 * v = 0.
- For all a in F: a * 0 = 0.
- For all a in F, v in V: (-a) * v = -(a * v), especially (-1) * v = -v.
Note: The above vector space axioms do not require the existence of multiplicative inverses guaranteed by F being a field. In fact, replacing F with a ring with unity generates consistent axioms not for a vector space, but for a module. However, most other definitions on vector spaces require F to be a field. Especially the definition of basis, and thus the definition of vector space dimension, requires an underlying scalar field.