Linear map

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 $V\to W$ 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.

"Linear transformation" redirects here. For fractional linear transformations, see Möbius transformation.

If a linear map is a bijection then it is called a linear isomorphism. In the case where $V=W$ , 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 $V$ and $W$ are real vector spaces (not necessarily with $V=W$ ), or it can be used to emphasize that $V$ 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 $V$ to $W$ always maps the origin of $V$ to the origin of $W$ . Moreover, it maps linear subspaces in $V$ onto linear subspaces in $W$ (possibly of a lower dimension);^[3] for example, it maps a plane through the origin in $V$ to either a plane through the origin in $W$ , a line through the origin in $W$ , or just the origin in $W$ . 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.

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

Additivity

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

Homogeneity

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

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] $f(c_{1}\mathbf {u} _{1}+\cdots +c_{n}\mathbf {u} _{n})=c_{1}f(\mathbf {u} _{1})+\cdots +c_{n}f(\mathbf {u} _{n}).$ Thus a linear map is one which preserves linear combinations.

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

A linear map $V\to K$ with $K$ viewed as a one-dimensional vector space over itself is called a linear functional.^[6]

These statements generalize to any left-module ${\textstyle {}_{R}M}$ over a ring $R$ without modification, and to any right-module upon reversing of the scalar multiplication.

A prototypical example that gives linear maps their name is a function $f:\mathbb {R} \to \mathbb {R} :x\mapsto cx$ , of which the is a line through the origin.^[7]

graph

More generally, any ${\textstyle \mathbf {v} \mapsto c\mathbf {v} }$ centered in the origin of a vector space is a linear map (here $c$ is a scalar).

homothety

The zero map ${\textstyle \mathbf {x} \mapsto \mathbf {0} }$ between two vector spaces (over the same ) is linear.

field

The on any module is a linear operator.

identity map

For real numbers, the map ${\textstyle x\mapsto x^{2}}$ is not linear.

For real numbers, the map ${\textstyle x\mapsto x+1}$ is not linear (but is an ).

affine transformation

If $A$ is a $m\times n$ , then $A$ defines a linear map from $\mathbb {R} ^{n}$ to $\mathbb {R} ^{m}$ by sending a column vector $\mathbf {x} \in \mathbb {R} ^{n}$ to the column vector $A\mathbf {x} \in \mathbb {R} ^{m}$ . Conversely, any linear map between finite-dimensional vector spaces can be represented in this manner; see the § Matrices, below.

real matrix

If ${\textstyle f:V\to W}$ is an between real normed spaces such that ${\textstyle f(0)=0}$ then $f$ is a linear map. This result is not necessarily true for complex normed space.^[8]

isometry

defines a linear map from the space of all differentiable functions to the space of all functions. It also defines a linear operator on the space of all smooth functions (a linear operator is a linear endomorphism, that is, a linear map with the same domain and codomain). Indeed, ${\frac {d}{dx}}\left(af(x)+bg(x)\right)=a{\frac {df(x)}{dx}}+b{\frac {dg(x)}{dx}}.$

Differentiation

A definite over some interval $I$ is a linear map from the space of all real-valued integrable functions on $I$ to $\mathbb {R}$ . Indeed, $\int _{u}^{v}\left(af(x)+bg(x)\right)dx=a\int _{u}^{v}f(x)dx+b\int _{u}^{v}g(x)dx.$

integral

An indefinite (or antiderivative) with a fixed integration starting point defines a linear map from the space of all real-valued integrable functions on $\mathbb {R}$ to the space of all real-valued, differentiable functions on $\mathbb {R}$ . Without a fixed starting point, the antiderivative maps to the quotient space of the differentiable functions by the linear space of constant functions.

integral

If $V$ and $W$ are finite-dimensional vector spaces over a field $F$ , of respective dimensions $m$ and $n$ , then the function that maps linear maps ${\textstyle f:V\to W}$ to $n \times m$ matrices in the way described in (below) is a linear map, and even a linear isomorphism.

§ Matrices

The of a random variable (which is in fact a function, and as such an element of a vector space) is linear, as for random variables $X$ and $Y$ we have $E[X+Y]=E[X]+E[Y]$ and $E[aX]=aE[X]$ , but the variance of a random variable is not linear.

expected value

rotation

reflection

by 2 in all directions: $\mathbf {A} ={\begin{pmatrix}2&0\\0&2\end{pmatrix}}=2\mathbf {I}$

scaling

: $\mathbf {A} ={\begin{pmatrix}1&m\\0&1\end{pmatrix}}$

horizontal shear mapping

skew of the y axis by an angle θ: $\mathbf {A} ={\begin{pmatrix}1&-\sin \theta \\0&\cos \theta \end{pmatrix}}$

: $\mathbf {A} ={\begin{pmatrix}k&0\\0&{\frac {1}{k}}\end{pmatrix}}$

squeeze mapping

onto the y axis: $\mathbf {A} ={\begin{pmatrix}0&0\\0&1\end{pmatrix}}.$

projection

the kernel is the space of solutions to the homogeneous equation f(v) = 0, and its dimension is the number of in the space of solutions, if it is not empty;

degrees of freedom

the co-kernel is the space of that the solutions must satisfy, and its dimension is the maximal number of independent constraints.

constraints

If, for some positive integer $n$ , the $n$ -th iterate of $T$ , $T n$ , is identically zero, then $T$ is said to be .

nilpotent

If $T 2 = T$ , then $T$ is said to be

idempotent

If $T = kI$ , where $k$ is some scalar, then $T$ is said to be a scaling transformation or scalar multiplication map; see .

scalar matrix

Applications[edit]

A specific application of linear maps is for geometric transformations, such as those performed in computer graphics, where the translation, rotation and scaling of 2D or 3D objects is performed by the use of a transformation matrix. Linear mappings also are used as a mechanism for describing change: for example in calculus correspond to derivatives; or in relativity, used as a device to keep track of the local transformations of reference frames.

Another application of these transformations is in compiler optimizations of nested-loop code, and in parallelizing compiler techniques.

– Z-module homomorphism

Additive map

– Conjugate homogeneous additive map

Antilinear map

– Special type of Boolean function

Bent function

– Linear transformation between topological vector spaces

Bounded operator

– Functional equation

Cauchy's functional equation

Continuous linear operator

– Linear map from a vector space to its field of scalars

Linear functional

– Distance-preserving mathematical transformation

Linear isometry

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Linear map

Additivity

Homogeneity

graph

homothety

field

identity map

affine transformation

real matrix

isometry

Differentiation

integral

integral

§ Matrices

expected value

scaling

horizontal shear mapping

squeeze mapping

projection

degrees of freedom

constraints

nilpotent

idempotent

scalar matrix

Applications[edit]

Additive map

Antilinear map

Bent function

Bounded operator

Cauchy's functional equation

Continuous linear operator

Linear functional

Linear isometry

Axler, Sheldon Jay

Handbook of Mathematics

Halmos, Paul Richard

Cambridge University Press

Katznelson, Yitzhak

ISBN

Lang, Serge

Rudin, Walter

Rudin, Walter

Rudin, Walter

Schaefer, Helmut H.

Schechter, Eric

ISBN

Tu, Loring W.

Wilansky, Albert