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Abelian Group

Basis

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.

Dimension

Field

A field F is an integral domain in which every non-zero element has a multiplicative inverse.

• The non-zero elements of F form a commutative group

Group

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.

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

Norm

• f(0) = 0
• f(x) > 0 for all x != 0
• f(cx) = |c| f(x)
• Triangle Inequality: f(x+yy) <= f(x) + f(y)

Ring

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)

Subspace

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. 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.