# Linear function
A linear function, in linear algebra, is a function between two vector spaces that preserves vector addition and scalar multiplication; it is also called a linear map, linear transformation, or, in some contexts, a homomorphism of vector spaces.[1](https://www.homepages.ucl.ac.uk/~ucahmto/0005_2021/Ch4.S14.html) • [2](https://math.ucdavis.edu/~linear/linear-guest.pdf)
| Key fact | Detail |
|---|---|
| Defining conditions | Additivity: \( f(\mathbf{x}+\mathbf{y}) = f(\mathbf{x})+f(\mathbf{y}) \); homogeneity: \( f(\lambda \mathbf{x}) = \lambda f(\mathbf{x}) \), for all vectors and all scalars \( \lambda \) in the field.[3](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html) |
| Consequence | Every linear map sends the zero vector to the zero vector, and \( f(\lambda_1 \mathbf{x}_1 + \cdots + \lambda_n \mathbf{x}_n) = \lambda_1 f(\mathbf{x}_1) + \cdots + \lambda_n f(\mathbf{x}_n) \).[3](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html) |
| Matrix representation | A linear map between finite-dimensional spaces is completely determined by its values on a basis; these values form an \( m \times n \) matrix depending on the chosen bases.[3](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html) |
| Rank–nullity | For a linear function \( f : V \to W \), \( \operatorname{rank}(f) + \operatorname{nullity}(f) = \dim V \).[4](https://www3.nd.edu/~math/restricted/CourseArchive/400Level/468/4681998S/Misc/linalg.pdf) |
| Injectivity test | A linear map is one-to-one if and only if its kernel is \( \{\mathbf{0}\} \).[3](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html) |
| Dimension constraint | A surjective linear map from \( X \) to \( Y \) implies \( \dim X \geq \dim Y \).[1](https://www.homepages.ucl.ac.uk/~ucahmto/0005_2021/Ch4.S14.html) |
| History | Peano gave the axiomatic definition of a real vector space and defined the concept of a linear transformation in 1888.[5](https://math.hawaii.edu/~lee/algebra/history.html) |
## Definition
Let \( V \) and \( W \) be vector spaces over the same field \( \mathbb{F} \). A function \( f : V \to W \) is linear if it satisfies two conditions:[3](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html)
1. **Additivity**: \( f(\mathbf{x}+\mathbf{y}) = f(\mathbf{x})+f(\mathbf{y}) \) for all \( \mathbf{x}, \mathbf{y} \in V \).
2. **Homogeneity**: \( f(\lambda \mathbf{x}) = \lambda f(\mathbf{x}) \) for all \( \mathbf{x} \in V \) and all \( \lambda \in \mathbb{F} \).
An equivalent single-condition statement is \( L(r\mathbf{u} + s\mathbf{v}) = rL(\mathbf{u}) + sL(\mathbf{v}) \) for all \( \mathbf{u}, \mathbf{v} \) and scalars \( r, s \); setting \( r = 1, s = 1 \) gives additivity and \( s = 0 \) gives homogeneity.[2](https://math.ucdavis.edu/~linear/linear-guest.pdf) The two conditions together mean the map respects the vector space structure, that is, scalar multiplication and vector addition.[1](https://www.homepages.ucl.ac.uk/~ucahmto/0005_2021/Ch4.S14.html)
**Names.** The concept is so common that it carries many names: linear map, linear function, linear transformation, and simply linear all mean the same thing.[1](https://www.homepages.ucl.ac.uk/~ucahmto/0005_2021/Ch4.S14.html) The word "homomorphism" is generally applied to functions from one kind of set to the same kind of set that respect the structures on those sets; linear maps are the homomorphisms of vector spaces.[2](https://math.ucdavis.edu/~linear/linear-guest.pdf) A map from a vector space to itself is often called a linear operator.[6](https://studyres.com/doc/1282605/linear-maps---people-pages---university-of-wisconsin?page=18) It is common to denote a linear function by a capital \( L \) as a reminder of its linearity.[2](https://math.ucdavis.edu/~linear/linear-guest.pdf)
## Immediate consequences
Because the two defining conditions can be applied repeatedly, a linear map preserves arbitrary linear combinations: \( f(\lambda_1 \mathbf{x}_1 + \cdots + \lambda_n \mathbf{x}_n) = \lambda_1 f(\mathbf{x}_1) + \cdots + \lambda_n f(\mathbf{x}_n) \) for any \( n \), scalars \( \lambda_i \), and vectors \( \mathbf{x}_i \).[3](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html) In words, the image of a sum is the sum of the images, the image of a scalar multiple is the scalar multiple of the image, and the image of a linear combination is the linear combination of the images.[7](https://people.math.binghamton.edu/mfochler/bu-mfx/bu-math-arv/math-330-arv/math-330-2022-08/html/lec-notes/bu-ln-330-m330-solns-2022-09-07.pdf)
Every linear map sends \( \mathbf{0}_U \) to \( \mathbf{0}_V \): from \( f(\mathbf{0}_U) = f(\mathbf{0}_U + \mathbf{0}_U) = f(\mathbf{0}_U) + f(\mathbf{0}_U) \), adding the [additive inverse](https://www.edgepedia.org/additive-inverse) of \( f(\mathbf{0}_U) \) to both sides gives \( \mathbf{0}_V = f(\mathbf{0}_U) \).[3](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html) This is why a translation such as \( \mathbf{x} \mapsto \mathbf{x} + \mathbf{b} \) with \( \mathbf{b} \neq \mathbf{0} \) is not linear: it moves the origin.[8](https://locusmath.org/books/lay-linear-algebra/2-7)
**Economy of specification.** A general real function of one variable requires one output to be specified for each input, an infinite amount of information. A linear function, even on an infinite domain, is specified by a small amount of information: knowing \( L \) on a single vector determines \( L \) on all its scalar multiples, and knowing \( L \) on a basis determines it everywhere.[2](https://math.ucdavis.edu/~linear/linear-guest.pdf)
## Examples
- The identity map \( \mathrm{id}_U : U \to U \) and the zero map sending every element of \( U \) to \( \mathbf{0}_V \) are linear for any vector spaces \( U, V \).[3](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html)
- For an \( m \times n \) matrix \( A \) over \( \mathbb{F} \), the map \( T_A : \mathbb{F}^n \to \mathbb{F}^m \) defined by \( T_A(\mathbf{x}) = A\mathbf{x} \) is linear; both defining conditions follow from properties of matrix multiplication and addition.[1](https://www.homepages.ucl.ac.uk/~ucahmto/0005_2021/Ch4.S14.html)
- The map \( T : \mathbb{R}^n \to \mathbb{R} \) sending \( (x_1, \ldots, x_n) \) to \( \sum_{i=1}^{n} x_i \) is linear, as is \( f(a,b,c) = a + b - 2c \).[1](https://www.homepages.ucl.ac.uk/~ucahmto/0005_2021/Ch4.S14.html) • [3](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html)
- Differentiation is linear: \( D(f) = \frac{df}{dx} \) is a linear map on the vector space of polynomials in one variable.[1](https://www.homepages.ucl.ac.uk/~ucahmto/0005_2021/Ch4.S14.html) • [3](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html)
- Non-examples: \( T(A) = A^2 \) on \( n \times n \) real matrices is not linear, and \( f(a,b,c) = \sqrt{a^2+b^2+c^2} \) is not linear, since for any nonzero \( \mathbf{x} \) one has \( f(-\mathbf{x}) \neq -f(\mathbf{x}) \).[1](https://www.homepages.ucl.ac.uk/~ucahmto/0005_2021/Ch4.S14.html) • [3](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html)
A linear functional, or linear form, is a linear map whose codomain is the field \( F \) itself, regarded as a one-dimensional vector space over itself.[9](https://algebrica.org/linear-maps/)
## Kernel, image, and the rank–nullity theorem
For a linear map \( f : U \to V \), the kernel (or null space) is \( \ker f = \{ \mathbf{u} \in U : f(\mathbf{u}) = \mathbf{0}_V \} \) and the image is \( \operatorname{im} f = \{ f(\mathbf{u}) : \mathbf{u} \in U \} \).[3](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html) Both are subspaces: the kernel is a subspace of \( U \) and the image is a subspace of \( V \), each closed under addition and scalar multiplication.[3](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html)
The kernel detects injectivity: \( f \) is one-to-one if and only if \( \ker f = \{\mathbf{0}_U\} \).[3](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html) The image detects surjectivity: \( f \) is onto exactly when its image equals \( W \).[4](https://www3.nd.edu/~math/restricted/CourseArchive/400Level/468/4681998S/Misc/linalg.pdf) The two dimensions are tied together by the rank–nullity theorem, \( \operatorname{rank}(f) + \operatorname{nullity}(f) = \dim V \).[4](https://www3.nd.edu/~math/restricted/CourseArchive/400Level/468/4681998S/Misc/linalg.pdf) [Dimension](https://www.edgepedia.org/dimension) also constrains existence of maps: a surjective linear map from \( X \) to \( Y \) forces \( \dim X \geq \dim Y \), the vector-space analog of the fact that an onto function between finite sets requires the domain to be at least as large as the codomain.[1](https://www.homepages.ucl.ac.uk/~ucahmto/0005_2021/Ch4.S14.html)
A linear map that is a bijection is called an isomorphism of vector spaces; if such a map exists from \( U \) to \( V \), the spaces are isomorphic, written \( U \cong V \).[3](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html) Every finite-dimensional vector space \( V \) is isomorphic to \( \mathbb{R}^n \), where \( n \) is the dimension of \( V \).[10](https://math.hws.edu/eck/math204/guide2020/12-linear-maps.html)
## Algebra of linear maps
Linear maps can be combined and remain linear. If \( f, g : U \to V \) are linear and \( \lambda \) is a scalar, the maps \( \lambda f \) and \( f + g \), defined pointwise, are linear; more generally \( \lambda f + \mu g \) is linear for scalars \( \lambda, \mu \).[3](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html) Composition preserves linearity as well: if \( f : U \to V \) and \( g : V \to W \) are linear, so is \( g \circ f : U \to W \).[3](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html) These operations make the set of linear maps from \( U \) to \( V \) into a vector space in its own right.[3](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html)
## Matrices and linear maps
For finite-dimensional spaces, linear maps and matrices are two views of the same object. A linear map \( f : U \to V \) is completely determined by a basis \( \mathcal{B} = \mathbf{b}_1, \ldots, \mathbf{b}_n \) of \( U \), a basis \( \mathcal{C} = \mathbf{c}_1, \ldots, \mathbf{c}_m \) of \( V \), and the scalars expressing each \( f(\mathbf{b}_j) \) in terms of the \( \mathbf{c}_i \); organizing these \( mn \) scalars into a rectangular grid gives the matrix \( [f]_{\mathcal{C}}^{\mathcal{B}} \), an \( m \times n \) matrix that depends on the choice of bases.[3](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html) Conversely, every matrix \( A \) defines a linear map \( \mathbf{x} \mapsto A\mathbf{x} \).[1](https://www.homepages.ucl.ac.uk/~ucahmto/0005_2021/Ch4.S14.html) Composition of maps corresponds to matrix multiplication, so a sequence of transformations can be combined into a single matrix before it is applied.[11](https://www.cs.bu.edu/fac/snyder/cs132-book/L13ComputerGraphics-Spring2021.html)
For a linear map \( f : V \to V \), an eigenvector is a nonzero \( \mathbf{v} \) with \( f(\mathbf{v}) = \lambda \mathbf{v} \) for some scalar \( \lambda \), the eigenvalue.[3](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html)
## Linear versus affine
In elementary contexts "linear function" often means a function of the form \( f(x) = mx + b \), whose graph is a line. [Computer graphics](https://www.edgepedia.org/computer-graphics) relies on the distinction: rotation, scaling, reflection, and shearing are linear transformations representable by matrices, but translation shifts the origin and is not linear. The standard fix is homogeneous coordinates, in which a point \( (x, y) \) is stored as \( (x, y, 1) \); in the larger space translation becomes a shear, which is linear, so a \( 3 \times 3 \) matrix (or \( 4 \times 4 \) in three dimensions) performs translation together with rotation, scaling, and reflection.[8](https://locusmath.org/books/lay-linear-algebra/2-7) • [12](https://math.hws.edu/eck/cs424/graphicsbook-1.0/c3/s5.html) Composing such operations then reduces to multiplying their matrices, which simplifies high-speed graphics hardware.[11](https://www.cs.bu.edu/fac/snyder/cs132-book/L13ComputerGraphics-Spring2021.html)
## History
The axiomatic framework for linear maps is comparatively recent. In 1888, Peano supplied axioms for real vector spaces and introduced the notion of a linear map from one vector space to another.[5](https://math.hawaii.edu/~lee/algebra/history.html) Linear transformations were listed among the core topics of a stand-alone linear algebra course by the Committee on the Undergraduate Program in [Mathematics](https://www.edgepedia.org/mathematics) of the Mathematical Association of America in 1965, alongside linear systems, matrices, vectors, and unitary geometry with characteristic values.[13](https://linear.sdsu.edu/Preface.html)
## References
1. [4.14 Linear maps, MATH0005 Algebra 1, UCL](https://www.homepages.ucl.ac.uk/~ucahmto/0005_2021/Ch4.S14.html)
2. [Linear Algebra, UC Davis (linear-guest.pdf)](https://math.ucdavis.edu/~linear/linear-guest.pdf)
3. [3.3 Linear maps, MATH0007: Algebra for Joint Honours Students, UCL](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html)
4. [Mathematics 468, University of Notre Dame](https://www3.nd.edu/~math/restricted/CourseArchive/400Level/468/4681998S/Misc/linalg.pdf)
5. [The Origins of Modern Algebra, University of Hawaii](https://math.hawaii.edu/~lee/algebra/history.html)
6. [Linear Maps, University of Wisconsin (studyres copy)](https://studyres.com/doc/1282605/linear-maps---people-pages---university-of-wisconsin?page=18)
7. [Math 330 Lecture Notes, Binghamton University](https://people.math.binghamton.edu/mfochler/bu-mfx/bu-math-arv/math-330-arv/math-330-2022-08/html/lec-notes/bu-ln-330-m330-solns-2022-09-07.pdf)
8. [Lay 2.7 Applications to computer graphics, Locus study guide](https://locusmath.org/books/lay-linear-algebra/2-7)
9. [Linear Maps, Algebrica](https://algebrica.org/linear-maps/)
10. [Isomorphisms and Homomorphisms, Hobart and William Smith](https://math.hws.edu/eck/math204/guide2020/12-linear-maps.html)
11. [Computer Graphics: Linear Algebra, Geometry, and Computation, Boston University](https://www.cs.bu.edu/fac/snyder/cs132-book/L13ComputerGraphics-Spring2021.html)
12. [Introduction to Computer Graphics, Section 3.5, Hobart and William Smith](https://math.hws.edu/eck/cs424/graphicsbook-1.0/c3/s5.html)
13. [Interactive Linear Algebra, San Diego State University](https://linear.sdsu.edu/Preface.html)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Linear and multilinear algebra › Vector spaces and linear maps*
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