In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, specifically in
linear algebraLinear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, the
coordinate space,
F^{n}, is the prototypical example of an
n-dimensional
vector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
over a
fieldIn abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
F. It can be defined as the product space of F over a finite
index setIn mathematics, the elements of a set A may be indexed or labeled by means of a set J that is on that account called an index set...
.
Definition
Let
F denote an arbitrary
fieldIn abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
(such as the
real numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s
R or the
complex numberA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s
C). For any positive
integerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
n, the space of all
n-tuples of elements of
F forms an
n-dimensional vector space over
F called
coordinate space and denoted
F^{n}.
An element of
F^{n} is written
where each
x_{i} is an element of
F. The operations on
F^{n} are defined by
The zero vector is given by
and the additive inverse of the vector
x is given by
Matrix notation
In standard
matrixIn mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
notation, each element of
F^{n} is typically written as a
column vector
and sometimes as a
row vector:
The coordinate space
F^{n} may then be interpreted as the space of all
n×1 column vectors, or all 1×
n row vectors with the ordinary matrix operations of addition and scalar multiplication.
Linear transformationIn mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...
s from
F^{n} to
F^{m} may then be written as
m×
n matrices which act on the elements of
F^{n} via left multiplication (when the elements of
F^{n} are column vectors) or right multiplication (when they are row vectors).
Standard basis
The coordinate space
F^{n} comes with a
standard basisIn mathematics, the standard basis for a Euclidean space consists of one unit vector pointing in the direction of each axis of the Cartesian coordinate system...
:
where 1 denotes the multiplicative identity in
F. To see that this is a basis, note that an arbitrary vector in
F^{n} can be written uniquely in the form
Discussion
It is a standard fact of linear algebra that every
n-dimensional vector space
V over
F is isomorphic to
F^{n}. It is a crucial point, however, that this isomorphism is not canonical. If it were, mathematicians would work only with
F^{n} rather than with abstract vector spaces.
A choice of isomorphism is equivalent to a choice of ordered basis for
V. To see this, let
- A : F^{n} → V
be a linear isomorphism. Define an ordered basis {
a_{i}} for
V by
- a_{i} = A(e_{i}) for 1 ≤ i ≤ n.
Conversely, given any ordered basis {
a_{i}} for
V define a linear map
A :
F^{n} →
V by
It is not hard to check that
A is an isomorphism. Thus ordered bases for
V are in 1-1 correspondence with linear isomorphisms
F^{n} →
V.
The reason for working with abstract vector spaces instead of
F^{n} is that it is often preferable to work in a
coordinate-free manner, i.e. without choosing a preferred basis. Indeed, many vector spaces that naturally show up in mathematics do not come with a preferred choice of basis.
It is possible and sometimes desirable to view a coordinate space
duallyIn mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often by means of an involution operation: if the dual of A is B, then the dual of B is A. As involutions sometimes have...
as the set of F-valued functions on a finite set; that is, each "point" of
F^{n} is viewed as a function whose domain is the finite set {1,2....n} and codomain
F. The function sends an element i of {1,2....n} to the value of the i'th coordinate of the "point", so
F^{n} is, dually, a set of functions.
See also
- real coordinate space, R^{n}
- complex coordinate space, C^{n}
- examples of vector spaces
This page lists some examples of vector spaces. See vector space for the definitions of terms used on this page. See also: dimension, basis.Notation. We will let F denote an arbitrary field such as the real numbers R or the complex numbers C...