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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 ${\displaystyle V\to W}$ between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism.

If a linear map is a bijection then it is called a linear isomorphism. In the case where ${\displaystyle V=W}$, a linear map is called a linear endomorphism. Sometimes the term linear operator refers to this case,^[1] but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that ${\displaystyle V}$ and ${\displaystyle W}$ are real vector spaces (not necessarily with ${\displaystyle V=W}$),^{[citation needed]} or it can be used to emphasize that ${\displaystyle V}$ is a function space, which is a common convention in functional analysis.^[2] Sometimes the term linear function has the same meaning as linear map, while in analysis it does not.

A linear map from ${\displaystyle V}$ to ${\displaystyle W}$ always maps the origin of ${\displaystyle V}$ to the origin of ${\displaystyle W}$. Moreover, it maps linear subspaces in ${\displaystyle V}$ onto linear subspaces in ${\displaystyle W}$ (possibly of a lower dimension);^[3] for example, it maps a plane through the origin in ${\displaystyle V}$ to either a plane through the origin in ${\displaystyle W}$, a line through the origin in ${\displaystyle W}$, or just the origin in ${\displaystyle W}$. Linear maps can often be represented as matrices, and simple examples include rotation and reflection linear transformations.

In the language of category theory, linear maps are the morphisms of vector spaces.

Definition and first consequences

Let ${\displaystyle V}$ and ${\displaystyle W}$ be vector spaces over the same field ${\displaystyle K}$. A function ${\displaystyle f:V\to W}$ is said to be a linear map if for any two vectors ${\textstyle \mathbf {u} ,\mathbf {v} \in V}$ and any scalar ${\displaystyle c\in K}$ the following two conditions are satisfied:

• Additivity / operation of addition
  $${\displaystyle f(\mathbf {u} +\mathbf {v} )=f(\mathbf {u} )+f(\mathbf {v} )}$$

  
• Homogeneity of degree 1 / operation of scalar multiplication
  $${\displaystyle f(c\mathbf {u} )=cf(\mathbf {u} )}$$

  

Thus, a linear map is said to be operation preserving. In other words, it does not matter whether the linear map is applied before (the right hand sides of the above examples) or after (the left hand sides of the examples) the operations of addition and scalar multiplication.

By the associativity of the addition operation denoted as +, for any vectors ${\textstyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{n}\in V}$ and scalars ${\textstyle c_{1},\ldots ,c_{n}\in K,}$ the following equality holds:^[4][5]

$${\displaystyle f(c_{1}\mathbf {u} _{1}+\cdots +c_{n}\mathbf {u} _{n})=c_{1}f(\mathbf {u} _{1})+\cdots +c_{n}f(\mathbf {u} _{n}).}$$

Denoting the zero elements of the vector spaces ${\displaystyle V}$ and ${\displaystyle W}$ by ${\textstyle \mathbf {0} _{V}}$ and ${\textstyle \mathbf {0} _{W}}$ respectively, it follows that ${\textstyle f(\mathbf {0} _{V})=\mathbf {0} _{W}.}$ Let ${\displaystyle c=0}$ and ${\textstyle \mathbf {v} \in V}$ in the equation for homogeneity of degree 1:

$${\displaystyle f(\mathbf {0} _{V})=f(0\mathbf {v} )=0f(\mathbf {v} )=\mathbf {0} _{W}.}$$

A linear map ${\displaystyle V\to K}$ with ${\displaystyle K}$Zdroj:https://en.wikipedia.org?pojem=Linear_transformation
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