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

- All bases for a subspace S contain the same number of vectors. This number is the dimension of S.

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.

- The characteristic of an integral domain or field is always either 0 or a prime number.
- A field with characteristic 0 is infinite but not all infinite fields have 0 characteristic.
- The integers, rationals, reals and complex numbers all have characteristic 0.
- If an integral domain has characteristic
*p*!=0, then (*x*+*y*)^{p}=*x*+^{p}*y*for all^{p}*x*,*y*. - If
*x*is a member of a field with characteristic 2, then*x*= -*x*. For all other fields*x*= -*x*implies*x*= 0.

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.

- dim(S) equals the maximum number of vectors in a linearly independent set taken from S.
- dim(S) equals the minimum number of vectors in a set that spans S.

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:

**Reflexive:****x**~**x**for all**x****Symmetric:****x**~**y**iff**y**~**x****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.

- The non-zero elements of
*F*form a commutative group. - If
*x*is a member of a field with characteristic 2, then*x*= -*x*. For all other fields*x*= -*x*implies*x*= 0. - The rationals, reals and complex numbers are all fields.

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

- The number of elements in a Galois field is always of the form
*p*where^{n}*p*is a prime and*n*a positive integer. - To within an isomporhpism, there is precisely one Galois field of order
*p*.^{n} - The characteristic of
GF(
*p*) equals^{n}*p*. - For prime
*p, GP*(*p*) is obtained by using the normal rules of arithmetic on the integers modulo*p*.

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

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

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

*Positive*: If**x**=0, then <**x**,**x**> =0 else <**x**,**x**> > 0*Additive*: <**x**+**y**,**z**> = <**x**,**z**> + <**y**,**z**>*Homogeneous*: ,*c***x**,**y**> =*c*<**x**,**y**> for all scalars c*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.

- Every inner product defines a vector
norm as ||
**x**|| = sqrt (<**x**,**x**>)- A vector norm may be derived from an inner product iff it satisfies the
parallelogram identity:
||
**x**+**y**||^{2}+||**x**-**y**||^{2}=2||**x**||^{2}+2||**y**||^{2} - If ||
**x**|| is derived from <**x**,**y**> then 4Re(<**x**,**y**>) = ||**x**+**y**||^{2}-||**x**-**y**||^{2}= 2||**x**+**y**||^{2}-||**x**||^{2}-||**y**||^{2}

- A vector norm may be derived from an inner product iff it satisfies the
parallelogram identity:
||
- The
*euclidean inner product*is <**x**,**y**> =**y**^{H}**x**- The
*euclidean norm*is ||**x**|| = sqrt (**x**^{H}**x**)

- The

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

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

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

- f(0) = 0
- f(
**x**) > 0 for all**x**!= 0 - f(
*c***x**) = |*c*| f(**x**) - Triangle Inequality: f(
**x**+**y**) <= f(**x**) + f(**y**)

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

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** = *c***y** for some non-zero
scalar *c*, then the equivalence classes are the elements of P(V).

- We define RP
^{n}= P(R^{n+1}) and CP^{n}= P(C^{n+1}) - If V has dimension
*n*+1, P(V) has dimension n. P(V) is not itself a vector space but it can be viewed as the union of (i) a vector space of dimension*n*and (ii) a set of "ideal points" of dimension*n*-1 [see 3 below]. - The elements of P(V) are the homogeneous vectors of V.
- The elements of RP
^{n}can be visualized in several ways:**Rays:**Lines through the origin in R^{n+1}. These are precisely the one-dimensional subspaces of R^{n+1}.**Hypersphere:**Pairs of antipodal points on the unit sphere in R^{n+1}. These are obtained by taking the equivalence class elements having unit length.**Augmented Affine Plane:**The union of (i) Points in the affine subspace that is defined by*x*_{n}_{+1}=1 and that is isomporphic to R^{n}and (ii) a set of "ideal points" (or "points at infinity") isomorphic to RP^{n+1}having that*x*_{n}_{+1}=0. Each of the ideal points corresponds to a "direction" in the subspace.- For n=1, we have the union of (i) a line isomrphic to R
^{1}and (ii) a single point at infinity. - For n=2, we have the union of (i) a plane isomorphic to R
^{2}and (ii) a circle at infinity in which opposite points are identified..

- For n=1, we have the union of (i) a line isomrphic to R

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

- 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.
- Closure: a*b is in
*R*for all a and b in*R* - Associativity: a*(b*c) = (a*b)*c for all a, b, c in
*R*(thus a*b*c is unambiguous) - * 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
*a***u**+*b***v**+,,,+*f***z** 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 *a***x**+*b***y** for any
*a* and *b* in F.

- The largest subspace is the entire vector space V.
- The smallest subspace consists of the single vector
**0**.

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:

- Closure:
*a***x**is an element of V - Distributivity:
*a*(**x**+**y**) =*a***x**+*a***y** - Distributivity: (
*a*+*b*)**x**=*a***x**+*b***x** - Associativity: (
*ab*)**x**=*a*(*b***x**) - Identity: 1
**x**=**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.

This page is part of The Matrix Reference Manual. Copyright © 1998-2022 Mike Brookes, Imperial College, London, UK. See the file gfl.html for copying instructions. Please send any comments or suggestions to "mike.brookes" at "imperial.ac.uk".

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