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A map is a function, as in the association of any of the four colored shapes in X to its color in Y. In mathematics, a map or mapping is a function in its general sense. [1] These terms may have originated as from the process of making a geographical map: mapping the Earth surface to a sheet of paper. [2]
The maps to symbol, ↦, is a rightward arrow protruding from a vertical bar. It is used in mathematics and in computer science to denote functions . In Z notation , a specification language used in software development, [ 1 ] this symbol is called the maplet arrow and the expression x ↦ y is called a maplet .
A multilinear map of one variable is a linear map, and of two variables is a bilinear map. More generally, for any nonnegative integer , a multilinear map of k variables is called a k-linear map. If the codomain of a multilinear map is the field of scalars, it is called a multilinear form.
This notation has also been used for other variants of floor and ceiling functions. 4. Iverson bracket: if P is a predicate, [] may denote the Iverson bracket, that is the function that takes the value 1 for the values of the free variables in P for which P is true, and takes the value 0 otherwise.
The map in question could be denoted (,) using the arrow notation. The expression x ↦ f ( x , t 0 ) {\displaystyle x\mapsto f(x,t_{0})} (read: "the map taking x to f of x comma t nought") represents this new function with just one argument, whereas the expression f ( x 0 , t 0 ) refers to the value of the function f at the point ( x 0 , t 0 ) .
At the same time, the mapping of a function to the value of the function at a point is a functional; here, is a parameter. Provided that f {\displaystyle f} is a linear function from a vector space to the underlying scalar field, the above linear maps are dual to each other, and in functional analysis both are called linear functionals .
Given a map :, the mapping cylinder is a space , together with a cofibration ~: and a surjective homotopy equivalence (indeed, Y is a deformation retract of ), such that the composition equals f. Thus the space Y gets replaced with a homotopy equivalent space M f {\displaystyle M_{f}} , and the map f with a lifted map f ~ {\displaystyle {\tilde ...
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication.