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In mathematics, specifically in linear algebra, the coordinate space, Fn, is the prototypical example of an n-dimensional vector space over a field F. Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ...
Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ...
The fundamental concept in linear algebra is that of a vector space or linear space. ...
In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. ...
Definition
Let F denote an arbitrary field (such as the real numbers R or the complex numbers C). For any positive integer n, the space of all n-tuples of elements of F forms an n-dimensional vector space over F called coordinate space and denoted Fn. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. ...
Please refer to Real vs. ...
The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...
In common usage positive is sometimes used in affirmation, as a synonym for yes or to express certainty. In mathematics, a number is called positive if it is bigger than zero. ...
The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ...
An element of Fn is written
where each xi is an element of F. The operations on Fn are defined by -
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The zero vector is given by and the additive inverse of the vector x is given by
Matrix notation In standard matrix notation the elements of Fn are written as column vectors For the square matrix section, see square matrix. ...
In linear algebra, a column vector is an m x 1 matrix, a matrix consisting of a single column. ...
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The coordinate space Fn may then be interpretted as the space of all n×1 column vectors with the ordinary matrix operations of addition and scalar multiplication.
Linear transformations from Fm to Fn may then be written as m×n matrices which act via left multiplication on the elements of Fn. In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...
In a similiar manner, the elements of the dual space (Fn)* are written as row vectors, so the dual space may be interpretted as the space of all 1×n row vectors. In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ...
In linear algebra, a row vector is a 1 x n matrix, a matrix consisting of a single row. ...
Standard basis The coordinate space Fn comes with a standard basis: In linear algebra, the standard basis for an -dimensional vector space is the basis obtained by taking the basis vectors where is the vector with a in the th coordinate and elsewhere. ...
where 1 denotes the multiplicative identity in F. To see that this is a basis, note that an arbitrary vector in Fn can be written uniquely in the form
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