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See all on Wikipediahttps://en.wikipedia.org/wiki/Linear_mapIn 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. The same … See more
Let and be vector spaces over the same field . A function is said to be a linear map if for any two vectors and any scalar the following two conditions are satisfied:
• Additivity / operation of addition f ( u + v ) = f ( u ) + f ( v ) … See moreThe composition of linear maps is linear: if and are linear, then so is their composition . It follows from this that the class of all vector spaces over a given field K, together with K-linear maps as morphisms, forms a category.
The See moreNo classification of linear maps could be exhaustive. The following incomplete list enumerates some important classifications that do not require any additional structure on the vector space.
Let V and W denote vector spaces over a field F and let T: V … See more
If and are finite-dimensional vector spaces and a basis is defined for each vector space, then every linear map from to can be represented by a matrix. This is useful because it allows … See more
A subtler invariant of a linear transformation is the cokernel, which is defined as
This is the dual notion to the kernel: just as the kernel is a subspace of the domain, the co-kernel is a quotient space of the target. Formally, … See moreGiven a linear map which is an endomorphism whose matrix is A, in the basis B of the space it transforms vector coordinates [u] as [v] = A[u]. As vectors change with the … See more
Wikipedia text under CC-BY-SA license- https://simple.wikipedia.org/wiki/Linear_mapping
In mathematics (particularly in linear algebra), a linear mapping (or linear transformation) is a mapping f between vector spaces that preserves addition and scalar multiplication. Definition. …
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- https://en.wikipedia.org/wiki/Transpose_of_a_linear_map
In linear algebra, the transpose of a linear map between two vector spaces, defined over the same field, is an induced map between the dual spaces of the two vector spaces. The transpose or algebraic adjoint of a linear map is often used to study the original linear map. This concept is generalised by adjoint functors.
Wikipedia · Text under CC-BY-SA license- Estimated Reading Time: 6 mins
- https://en.wikipedia.org/wiki/Linear_function
- In mathematics, the term linear function refers to two distinct but related notions: In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For distinguishing such a linear function from the other concept, the term affine function is often used. In linear ...
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- https://en.wikipedia.org/wiki/Discontinuous_linear_map
- In mathematics, linear maps form an important class of "simple" functions which preserve the algebraic structure of linear spaces and are often used as approximations to more general functions. If the spaces involved are also topological spaces, then it makes sense to ask whether all linear maps are continuous. It turns out that for maps defined on...
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- https://en.wikipedia.org/wiki/Linear_algebra
Linear algebra is the branch of mathematics concerning linear equations such as: + + =, linear maps such as: (, …,) + +,and their representations in vector spaces and through matrices.. Linear algebra is central to almost all areas of …
- https://en.wikipedia.org/wiki/Multilinear_map
In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function. where and are vector spaces …
Linear map - Wikipedia
https://wiki.alquds.edu/?query=Linear_mapMar 10, 2022 · These can be interpreted thus: given a linear equation f(v) = w to solve, the kernel is the space of solutions to the homogeneous equation f(v) = 0, and its dimension is the numbe
- https://en.wikipedia.org/wiki/Bilinear_map
In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is …
- https://en.wikipedia.org/wiki/Antilinear_map
In contrast, a linear map is a function that is additive and homogeneous, where is called homogeneous if f ( a x ) = a f ( x ) for all vectors x and all scalars a . {\displaystyle …
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