Website owner:  James Miller

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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

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