# Linear map A linear map (also called a linear transformation or linear mapping) is a function \( f: U \to V \) between vector spaces over the same field that preserves vector addition and scalar multiplication, so that \( f(\mathbf{u}+\mathbf{w}) = f(\mathbf{u})+f(\mathbf{w}) \) and \( f(\lambda \mathbf{u}) = \lambda f(\mathbf{u}) \) for all vectors and all scalars \( \lambda \)[1](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html). Equivalently, a single condition \( f(c\mathbf{v}+d\mathbf{w}) = c\,f(\mathbf{v})+d\,f(\mathbf{w}) \) captures both requirements[2](https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/4d876a9159e32543eb0d73b4d4382f4c_MIT18_06S10ZoomNotes.pdf). Linear maps are the structure-preserving functions of linear algebra, the way group homomorphisms preserve group structure, and they include rotations, dilations, projections, differentiation on suitable function spaces, and multiplication by a matrix[1](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html) • [3](https://textbooks.math.gatech.edu/ila/linear-transformations.html). | Key fact | Detail | |---|---| | Defining property | \( T(c\mathbf{v}+d\mathbf{w}) = c\,T(\mathbf{v})+d\,T(\mathbf{w}) \) for all vectors \( \mathbf{v}, \mathbf{w} \) and scalars \( c, d \); in particular \( T(\mathbf{0}) = \mathbf{0} \)[2](https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/4d876a9159e32543eb0d73b4d4382f4c_MIT18_06S10ZoomNotes.pdf) • [3](https://textbooks.math.gatech.edu/ila/linear-transformations.html) | | Kernel and image | \( \ker f = \{\mathbf{u} : f(\mathbf{u})=\mathbf{0}\} \) is a subspace of the domain; \( \operatorname{im} f = \{f(\mathbf{u})\} \) is a subspace of the codomain[1](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html) | | Rank-nullity theorem | For finite-dimensional \( U \), \( \dim \operatorname{im} f + \dim \ker f = \dim U \)[1](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html) | | Injectivity test | A linear map is one-to-one if and only if \( \ker f = \{\mathbf{0}\} \)[1](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; with bases chosen, it is an \( m \times n \) matrix \( [f]^{\mathcal{B}}_{\mathcal{C}} \)[1](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html) | | Maps versus matrices | Linear maps \( \mathbb{R}^n \to \mathbb{R}^m \) correspond one-to-one to \( m \times n \) matrices via \( f(x)=Ax \); affine maps have the form \( f(x)=Ax+b \)[4](https://ecampusontario.pressbooks.pub/optimizationmodelsandapplications/chapter/linear-maps/) | | Historical anchor | Peano gave the axiomatic definition of a vector space and the concept of a linear transformation in 1888; Sylvester introduced the term "matrix" in 1850[5](https://math.hawaii.edu/~lee/algebra/history.html) • [6](https://people.math.harvard.edu/~knill/teaching/math21b2001/history/index.html) | ## Definition and immediate consequences The two preservation rules have consequences that follow directly. A linear map \( f \) always sends the zero vector of the domain to the zero vector of the codomain, because \( f(\mathbf{0}_U) = f(\mathbf{0}_U+\mathbf{0}_U) = f(\mathbf{0}_U)+f(\mathbf{0}_U) \), and subtracting \( f(\mathbf{0}_U) \) from both sides gives \( \mathbf{0}_V \)[1](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html). This gives a quick necessary test: any transformation that does not map zero to zero, such as a translation, is automatically not linear[3](https://textbooks.math.gatech.edu/ila/linear-transformations.html). The second consequence is the superposition principle: \( f \) applied to any linear combination equals the same combination of the images, \( f(x_1\mathbf{e}_1 + \cdots + x_n\mathbf{e}_n) = x_1 f(\mathbf{e}_1) + \cdots + x_n f(\mathbf{e}_n) \)[3](https://textbooks.math.gatech.edu/ila/linear-transformations.html) • [7](https://www.maths.tcd.ie/~vdots/teaching/files/MA1112-1819/1112L2handout.pdf). In engineering contexts this additivity-and-scaling behavior is itself called superposition[3](https://textbooks.math.gatech.edu/ila/linear-transformations.html). A coordinate test is useful in practice: if a transformation is defined by formulas in the coordinates, it is linear if and only if each coordinate is a linear expression in the variables with no constant term[3](https://textbooks.math.gatech.edu/ila/linear-transformations.html). ## Kernel, image, rank, and nullity Two subspaces encode everything about how a linear map behaves with respect to dimension[8](https://aleph0.clarku.edu/~ma130/ranknullity.pdf). - The **kernel** (or null space), \( \ker f = \{\mathbf{u} \in U : f(\mathbf{u}) = \mathbf{0}_V\} \), is the inverse image of the zero vector. It is a subspace of the domain[1](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html) • [8](https://aleph0.clarku.edu/~ma130/ranknullity.pdf). - The **image** (or range), \( \operatorname{im} f = \{f(\mathbf{u}) : \mathbf{u} \in U\} \), is the set of values of \( f \). It is a subspace of the codomain[1](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html) • [8](https://aleph0.clarku.edu/~ma130/ranknullity.pdf). The **rank** of \( f \) is \( \dim \operatorname{im} f \) and the **nullity** is \( \dim \ker f \)[1](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html). The rank-nullity theorem states that for a linear map \( f: U \to V \) with \( U \) finite-dimensional, \[ \dim \operatorname{im} f + \dim \ker f = \dim U. \] The proof extends a basis \( \mathbf{k}_1, \ldots, \mathbf{k}_m \) of the kernel to a basis \( \mathbf{k}_1, \ldots, \mathbf{k}_m, \mathbf{u}_1, \ldots, \mathbf{u}_n \) of \( U \); the images \( f(\mathbf{u}_1), \ldots, f(\mathbf{u}_n) \) then form a basis of the image, so \( \dim \operatorname{im} f = n \)[1](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html). The theorem is sometimes called the Fundamental Theorem of Linear Mappings[9](https://www.youtube.com/watch?v=bzITcRE5DpY), and one reading of it is that the smaller the kernel, the more faithfully the image captures the domain[10](https://web.williams.edu/Mathematics/lg5/250/LS33.pdf). The theorem yields the standard injectivity and surjectivity criteria. A linear map is one-to-one if and only if \( \ker f = \{\mathbf{0}_U\} \)[1](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html). When \( \dim U = \dim V \) and both are finite-dimensional, being bijective, being one-to-one, and being onto are all equivalent, because rank-nullity forces the image to fill \( V \) exactly when the kernel is trivial[1](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html). A bijective linear map is called an **isomorphism**, and spaces related by one are written \( U \cong V \)[1](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html). Kernels and images also behave predictably under composition: for composable maps \( V \xrightarrow{T} W \xrightarrow{U} X \), \( \ker(T) \subseteq \ker(U \circ T) \) and \( \operatorname{im}(U \circ T) \subseteq \operatorname{im}(U) \)[8](https://aleph0.clarku.edu/~ma130/ranknullity.pdf). ## Matrices and linear maps A linear map out of a finite-dimensional space is completely determined by its values on a basis. If \( \mathcal{B} = \mathbf{b}_1, \ldots, \mathbf{b}_n \) is a basis of \( U \) and \( \mathcal{C} = \mathbf{c}_1, \ldots, \mathbf{c}_m \) a basis of \( V \), writing each \( f(\mathbf{b}_j) = \sum_{i=1}^m a_{ij} \mathbf{c}_i \) produces an \( m \times n \) matrix \( [f]^{\mathcal{B}}_{\mathcal{C}} = (a_{ij}) \), whose \( j \)-th column records the coordinates of \( f(\mathbf{b}_j) \)[1](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html). Conversely, every matrix defines a linear map by \( \mathbf{v} \mapsto A\mathbf{v} \), so over \( \mathbb{R}^n \to \mathbb{R}^m \) there is a one-to-one correspondence between linear maps and matrices[4](https://ecampusontario.pressbooks.pub/optimizationmodelsandapplications/chapter/linear-maps/) • [3](https://textbooks.math.gatech.edu/ila/linear-transformations.html). In this correspondence the image of the map \( L_A \) is the column space of \( A \), the subspace generated by the columns, and the rank of the map is independent of the choice of basis[11](https://www.maths.tcd.ie/~mozgovoy/data/notes_LA2.pdf). The kernel corresponds to the solution space of the homogeneous system \( A\mathbf{x} = \mathbf{0} \), whose dimension equals the number of free variables, that is, the number of non-pivotal columns in the reduced row echelon form[7](https://www.maths.tcd.ie/~vdots/teaching/files/MA1112-1819/1112L2handout.pdf). ## Linear versus affine maps An **affine transformation** is a linear transformation followed by a translation: \( f(\mathbf{x}) = A\mathbf{x} + \mathbf{c} \)[12](https://math.hws.edu/eck/math204/guide2020/16-affine-maps.html). Every linear transformation is affine (with \( \mathbf{c} = \mathbf{0} \)), but an affine map with nonzero translation vector is not linear, since it does not map \( \mathbf{0} \) to \( \mathbf{0} \); an affine map is linear if and only if \( f(\mathbf{0}) = \mathbf{0} \)[13](https://mathweb.ucsd.edu/~sbuss/MathCG2/SecondEdDraft.pdf) • [12](https://math.hws.edu/eck/math204/guide2020/16-affine-maps.html). The one-dimensional function \( y = mx + b \) is therefore affine rather than linear in the linear-algebra sense[14](https://www.dca.ufrn.br/~lmarcos/courses/compgraf/notes/transform/tsld006.htm). The translation vector and the linear part of an affine map are uniquely determined, with \( \mathbf{u} = A(\mathbf{0}) \)[13](https://mathweb.ucsd.edu/~sbuss/MathCG2/SecondEdDraft.pdf). In computer graphics, affine maps of \( \mathbb{R}^3 \) are represented by \( 4 \times 4 \) matrices through homogeneous coordinates, appending an extra coordinate 1 so that the translation becomes part of a single matrix product; the composition of affine maps is then the ordinary matrix product[12](https://math.hws.edu/eck/math204/guide2020/16-affine-maps.html). Affine maps also serve as local approximations: if \( f: \mathbb{R}^n \to \mathbb{R}^m \) is differentiable, then near a point \( x_0 \) it is approximated by the affine map \( \tilde{f}(x) = f(x_0) + A(x - x_0) \), where \( A \) is the Jacobian matrix of partial derivatives \( A_{ij} = \partial f_i / \partial x_j \, (x_0) \)[4](https://ecampusontario.pressbooks.pub/optimizationmodelsandapplications/chapter/linear-maps/). ## Eigenvalues and endomorphisms For a linear map \( T: V \to V \) from a space to itself, an **eigenvector** is a nonzero \( \mathbf{v} \) with \( T(\mathbf{v}) = \lambda \mathbf{v} \) for some scalar \( \lambda \), the eigenvalue[1](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html). Eigenvectors with different eigenvalues are linearly independent[1](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html). The concept extends the matrix definition: a column vector is an eigenvector of the map \( T_A(\mathbf{v}) = A\mathbf{v} \) exactly when it is an eigenvector of \( A \)[1](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html). The framework reaches beyond column vectors. On the space of polynomials of degree at most \( n \), the differentiation map \( D(f) = df/dx \) is linear; every eigenvector of \( D \) on that space is a constant polynomial, so \( D \) has no basis of eigenvectors and is not diagonalizable there. On the space of differentiable functions \( \mathbb{R} \to \mathbb{R} \), the function \( f(x) = e^{\lambda x} \) is a \( \lambda \)-eigenvector of differentiation[1](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html). A map is **diagonalizable** when some basis of \( V \) consists of eigenvectors of \( T \)[1](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html). ## History The modern concept assembled slowly from older parts. Work on systems of linear equations dates to antiquity, with Chinese sources recording what is now called [Gaussian elimination](https://www.edgepedia.org/gaussian-elimination) around 200 BCE, and determinants were used by Seki and Leibniz in 1683[6](https://people.math.harvard.edu/~knill/teaching/math21b2001/history/index.html). Properties of matrices were considered before matrices were formally defined: Sylvester introduced the term "matrix" in 1850 as a shorthand for substitutions of variables, that is, what are now called linear transformations, and Cayley studied their composition in 1855[5](https://math.hawaii.edu/~lee/algebra/history.html) • [6](https://people.math.harvard.edu/~knill/teaching/math21b2001/history/index.html). Frobenius defined the rank of a matrix in 1878, and Peano in 1888 gave the axiomatic definition of a real vector space together with the concept of a linear transformation between vector spaces, the form in which the subject is taught today[5](https://math.hawaii.edu/~lee/algebra/history.html) • [6](https://people.math.harvard.edu/~knill/teaching/math21b2001/history/index.html). ## References 1. [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) 2. [ZoomNotes for Linear Algebra, MIT OpenCourseWare 18.06 (Spring 2010)](https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/4d876a9159e32543eb0d73b4d4382f4c_MIT18_06S10ZoomNotes.pdf) 3. [Linear Transformations, Interactive Linear Algebra, Georgia Tech](https://textbooks.math.gatech.edu/ila/linear-transformations.html) 4. [Linear Maps, Hyper-Textbook: Optimization Models and Applications](https://ecampusontario.pressbooks.pub/optimizationmodelsandapplications/chapter/linear-maps/) 5. [The Origins of Modern Algebra, University of Hawaii](https://math.hawaii.edu/~lee/algebra/history.html) 6. [Some history of linear algebra, Harvard Math 21b (O. Knill)](https://people.math.harvard.edu/~knill/teaching/math21b2001/history/index.html) 7. [MA1112: Linear Algebra II, Lecture 2, Trinity College Dublin (V. Dotsenko)](https://www.maths.tcd.ie/~vdots/teaching/files/MA1112-1819/1112L2handout.pdf) 8. [Kernel, image, nullity, and rank, Math 130, Clark University (D. Joyce)](https://aleph0.clarku.edu/~ma130/ranknullity.pdf) 9. [Kernel and Image, Rank-Nullity Theorem Explained (video)](https://www.youtube.com/watch?v=bzITcRE5DpY) 10. [Lecture 33: Rank-Nullity, Williams College](https://web.williams.edu/Mathematics/lg5/250/LS33.pdf) 11. [1.4. Rank and nullity, Trinity College Dublin notes](https://www.maths.tcd.ie/~mozgovoy/data/notes_LA2.pdf) 12. [Affine Transformations, Math 204 guide, Hobart and William Smith (D. Eck)](https://math.hws.edu/eck/math204/guide2020/16-affine-maps.html) 13. [Mathematics for Computer Graphics, 2nd edition draft (S. Buss), UCSD](https://mathweb.ucsd.edu/~sbuss/MathCG2/SecondEdDraft.pdf) 14. [Linear and Affine Maps, computer graphics course notes, UFRN](https://www.dca.ufrn.br/~lmarcos/courses/compgraf/notes/transform/tsld006.htm) --- *Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Linear and multilinear algebra › Vector spaces and linear maps* *Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026* *Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.* License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license