Algebraic Structures

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Abelian Group

See under Commutative Group.


A set of vectors u, v, ..., z form a basis for a subspace S iff they span S and are linearly independent.

Characteristic of an Integral domain or Field

All integral domains (and hence fields) contain at least two elements: 0 and 1. The characteristic of an integral domain is the minimum number of 1's that must be added together to equal 0. The characteristic equals zero if no sum of 1's ever equals 0.

Commutative or Abelian Group

A group having the operation + is commutative or Abelian if a+b=b+a for all a and b in G.


The dimension of a subspace S, dim(S), is the number of vectors in a basis for S.

Equivalence Relation

An equivalence relation is a relation that divides the set into disjoint classes and that satisfies the three axioms given below. We write x ~ y if x and y are in the same class, i.e. if the relation is true for the pair x and y. Axioms:

  1. Reflexive: x ~ x for all x
  2. Symmetric: x ~ y iff y ~ x
  3. Transitive: If x ~ y and y ~ z then x ~ z


A field F (also called a rational domain) is an integral domain in which every non-zero element has a multiplicative inverse.

Galois Field

A field with a finite number of elements, k, is a Galois Field and denoted by GF(k).


A group G is a set of elements a, b, c, ... with a binary operation + satisfying:

  1. Closure: a+b is in G for all a and b in G
  2. Associativity: a+(b+c) = (a+b)+c for all a, b, c in G (thus a+b+c is unambiguous)
  3. Identity: There exisits an identity element, e, in G such that e+a=a for all a in G
  4. Inverse: For each a in G, there exists an element b in G satisfying a+b=e.

Inner Product

An inner product is a scalar function defined on ordered pairs <x, y> of vectors that satisfies:

  1. Positive: If x=0, then <x,x> =0 else <x, x> > 0
  2. Additive: <x+y, z> = <x, z> + <y, z>
  3. Homogeneous: ,cx, y> = c <x, y> for all scalars c
  4. Hermitian: <x, y> = conj(<y, x>)

where "scalar" refers to the field associated with the vector space. Note that condition 4 implies that <x, x> is real.

Integral Domain

An integral domain, D, is a ring of elements a, b, c, ... in which:

  1. Identity: There is an identity element, 1, such that 1*a = a for all a in D.
  2. Commutativity: a*b = b*a for all a and b in D
  3. Cancellation: If a != 0, then a*b = a*c implies b = c.

Linear Independence

A set of vectors u, v, ..., z are linearly independent iff  au+bv+,,,+fz = 0 implies that a= b= ...= f =0.


A norm is is a real-valued function f(x) defined on a ring of objects satisfying the following:

Note that a matrix norm has to satisfy the additional condition ||XY|| <= ||X|| ||Y||

Projective Space

The projective space, P(V), of a vector space, V, consists of the set of all one-dimensional subspaces of V.
Alternatively, if we define an equivalence relation on the non-zero elements of V with x ~ y iff x = cy for some non-zero scalar c, then the equivalence classes are the elements of P(V).


A ring R is a set of elements a, b, c, ... with two binary operations + and * such that:

  1. R is a commutative group with respect to + in which the identity element is written 0 and the inverse of the element a is written -a.
  2. Closure: a*b is in R for all a and b in R
  3. Associativity: a*(b*c) = (a*b)*c for all a, b, c in R (thus a*b*c is unambiguous)
  4. * is Distributive over +: a*(b+c) = (a*b) + (a*c) and (a+b)*c = (a*c) + (b*c)


A set of vectors u, v, ..., z span a subspace S iff all members of S can be expressed in the form au+bv+,,,+fz where a, b, ..., f are scalars.


A subspace S of a vector space, V, over a field F is a subset of the vectors in V that satisfies the following closure property: if x and y are elements of  S then so is ax+by for any a and b in F.

Vector Space

A vector space, V, over a field F is a set of elements (called vectors) x, y, z, ... that form a commutative group under an operation + such that for any x, y in V and a, b in F:

  1. Closure: ax is an element of V
  2. Distributivity: a(x+y) = ax + ay
  3. Distributivity: (a+b)x = ax + bx
  4. Associativity: (ab)x = a(bx)
  5. Identity: 1x = x where 1 is the multiplicative identity in the field

Note that the symbol + is used both for the addition of vectors and for elements of the field and that the symbol * is used both for the product of a field element (scalar) and a vector as well as the product of two scalars. In practice this causes no confusion.

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