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Werg22
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How should I represent a vector space V with scalar field F and operation + and x? Is the notation [V, F, +, x] used, or should I use something else?
A vector space is a mathematical structure that consists of a set of vectors and operations that can be performed on those vectors. These operations include addition, scalar multiplication, and a zero vector.
The purpose of notation for representing vector spaces is to provide a concise and standardized way to express the elements and operations of a vector space. This notation allows for easy communication and manipulation of vector space concepts and calculations.
The most common symbols used in vector space notation include a bold lowercase letter for a vector (e.g. v), a bold uppercase letter for a matrix (e.g. A), and Greek letters for scalars (e.g. α). Other symbols may also be used to represent specific operations, such as the dot product (·) and cross product (×).
Vector spaces can be represented using matrices by choosing a basis for the vector space and expressing each vector in terms of that basis. The basis vectors are then written as columns in a matrix, with the coefficients of each vector written as entries in the corresponding column. This matrix is known as the coordinate matrix or representation matrix.
In vector space notation, a vector usually refers to a single element of the vector space, while a matrix refers to a collection of vectors. A vector can be thought of as a 1-dimensional matrix, while a matrix can be thought of as a higher-dimensional collection of vectors. Additionally, different operations may be performed on vectors and matrices in vector space notation.