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PROPERTIES THAT FOLLOW FROM THE AXIOMATIC DEFINITIONS OF DIFFERENT SPACES

In modern abstract mathematics properties are deduced from sets of axioms just as in Euclidean Geometry all results are arrived at through deduction from a set of axioms.

1. Linear Space. The following properties follow directly from the eight axioms that constitute the definition of a linear space:

- subspaces (the space may contain subspaces)

- basis vectors (if the space contains subspaces there will be

basis vectors that span the subspaces such that any vector in

a subspace can be expressed as a linear combination of a set

of basis vectors)

- dimension (the space has a dimension given by the number of

linearly independent vectors required to span it)

- coordinate representation of vectors (if e1, e2, .... , en are a set

of independent basis vectors for the space any vector v in the

space can be represented as

v = a1e1 + a2e2 + .... + anen

where a1, a2, .... , an are numbers from a number field F

which can be regarded as coordinates of the vector v with

respect to that basis)

2. Linear space with an inner product. A linear space with an inner product has all the properties of a linear space plus the following properties:

- orthogonality (concept of the orthogonality of one vector to

another vector)

- orthogonal bases

3. Linear space with an inner product and a norm. A linear space with both an inner product and a norm has all the properties of a linear space with an inner product plus:

- orthonormal bases

- coordinate representation with respect to orthonormal bases