1 Definition and basic properties
A linear transformation is a mapping between vector spaces that preserves the two fundamental operations of vector addition and scalar multiplication. This preservation makes linear transformations the basic morphisms of linear algebra, allowing algebraic structure to be carried from one vector space to another without distortion of linear relationships.
In practice, linear transformations include geometric actions such as rotations, reflections, projections, and scaling, as well as algebraic processes used to solve equations and analyze data. Their importance lies in the fact that many complicated problems can be recast in terms of linear structure.
1.1 Vector spaces and mappings
A linear transformation is defined on one vector space and takes values in another vector space, possibly over the same field. The domain and codomain may consist of ordinary coordinate vectors, polynomials, matrices, or other objects that satisfy the axioms of a vector space.
Not every function between vector spaces is linear. To qualify, the mapping must interact with the vector space operations in a specific way. This requirement distinguishes linear transformations from more general functions and gives them their characteristic usefulness in theory and computation.
1.2 Additivity and homogeneity
A transformation \(T\) is linear if it satisfies additivity and homogeneity. Additivity means that applying \(T\) to a sum gives the sum of the transformed vectors: \[ T(u+v)=T(u)+T(v). \] Homogeneity means that scalar multiplication is preserved: \[ T(cu)=cT(u). \]
Together, these properties imply that linear combinations behave predictably under the transformation. They ensure that the map respects the algebraic structure of the vector space rather than introducing nonlinear effects.
1.3 Equivalent formulations
The defining properties of linearity can be expressed in several equivalent ways. In many texts, a transformation is taken to be linear if it preserves all linear combinations, which compresses the additivity and homogeneity conditions into a single statement.
A useful consequence is that the action of a linear transformation is determined by its values on a spanning set. This makes it possible to describe the entire map from relatively little information.
1.3.1 Preservation of linear combinations
If a transformation is linear, then for any vectors \(v_1,\dots,v_k\) and scalars \(a_1,\dots,a_k\), \[ T(a_1v_1+\cdots+a_kv_k)=a_1T(v_1)+\cdots+a_kT(v_k). \] This property extends the two defining rules to any finite combination of vectors. It is one of the most practical ways to recognize and use linearity.
1.3.2 Mapping of the zero vector
Every linear transformation sends the zero vector to the zero vector. This follows from either additivity or homogeneity: \[ T(0)=0. \] The zero vector must remain fixed because it is the additive identity in every vector space. This fact is often used as a quick test for nonlinearity.
1.4 Examples of linear transformations
Common examples include the zero transformation, which sends every vector to zero, and the identity transformation, which leaves every vector unchanged. Another basic example is coordinate projection, which discards some components while retaining others.
Linear transformations also appear as differentiation on suitable spaces of polynomials or smooth functions, and as matrix multiplication by a fixed matrix. Each of these examples preserves the linear structure in a different but closely related way.
2 Matrix representation
Linear transformations between finite-dimensional vector spaces can be represented by matrices once bases are chosen. This connection is central to computational linear algebra, because it converts an abstract map into an array of numbers that can be manipulated by standard algebraic methods.
Matrix representation does not change the transformation itself; rather, it records how the transformation acts relative to chosen coordinates. Different bases produce different matrices for the same underlying map.
2.1 Matrices of linear transformations
If the domain has a basis, the image of each basis vector under a linear transformation determines the transformation completely. Writing those images in coordinates produces the columns of the associated matrix.
This construction is especially useful because it reduces questions about the transformation to questions about the matrix. Properties such as invertibility, rank, and eigenvalues can then be studied using matrix techniques.
2.2 Standard matrix
For linear transformations on spaces like \(\mathbb{R}^n\), the standard matrix is the matrix obtained using the standard basis. In that setting, the transformation of a vector is computed by ordinary matrix multiplication.
The standard matrix provides a direct bridge between geometric actions and algebraic calculation. Once it is known, the transformation can be applied efficiently to any coordinate vector in the standard basis.
2.3 Transformation under different bases
A single linear transformation has many matrix representations, depending on the chosen bases for the domain and codomain. Changing bases changes the coordinate description, even though the underlying map remains the same.
This flexibility is important when one basis makes the structure of the transformation easier to see, such as when seeking diagonal form or simplifying calculations in a particular application.
2.3.1 Change of basis
A change of basis replaces one coordinate system with another. The same vector is then described by different coordinate tuples, and the matrix of the transformation must be adjusted accordingly.
Change of basis matrices translate coordinates from one basis to another. They are used repeatedly in the study of linear operators and in simplifying computations by selecting more convenient coordinates.
2.3.2 Similarity transformations
When a linear operator is expressed in two different bases of the same space, the two matrices are related by similarity. Similar matrices represent the same transformation in different coordinate systems.
Similarity preserves many important properties, including determinant, trace, characteristic polynomial, and eigenvalues. This makes similarity a fundamental concept in understanding when two matrices encode the same linear structure.
2.4 Coordinate vectors and matrix multiplication
A vector in a finite-dimensional space can be represented by a coordinate vector relative to a basis. Applying a linear transformation then corresponds to multiplying that coordinate vector by the appropriate matrix.
This relationship explains why matrix multiplication is so central in linear algebra. It is not merely a formal operation; it encodes the composition of a linear map with the process of expressing vectors in coordinates.
3 Kernel and image
Two central subspaces associated with a linear transformation are its kernel and image. These subspaces measure, respectively, which vectors are sent to zero and which vectors can actually be reached as outputs.
Together, they describe much of the structure of the transformation. They are essential in determining whether the map is one-to-one, onto, or both.
3.1 Kernel (null space)
The kernel of a linear transformation consists of all vectors in the domain that map to the zero vector in the codomain. It is also called the null space.
The kernel reveals how much information is lost under the transformation. If the kernel contains only the zero vector, then no nonzero vector is collapsed to zero, and the transformation is injective.
3.2 Image (range)
The image of a linear transformation is the set of all vectors in the codomain that arise as outputs of the map. It is also called the range.
Because the image consists of all attainable outputs, it measures the extent to which the codomain is covered. The image is always a subspace of the codomain, reflecting the linearity of the transformation.
3.3 Rank and nullity
The dimension of the image is called the rank, and the dimension of the kernel is called the nullity. These numerical invariants summarize the size of the two principal subspaces attached to the transformation.
Rank and nullity are often used to compare domain and codomain dimensions and to diagnose whether a system of equations has unique, multiple, or no solutions.
3.3.1 Rank-nullity theorem
For a linear transformation from a finite-dimensional space, the sum of rank and nullity equals the dimension of the domain: \[ \text{rank} + \text{nullity} = \dim(\text{domain}). \] This theorem is one of the most important results in linear algebra. It expresses a balance between the degrees of freedom preserved by the map and those lost in the kernel.
3.4 Injectivity and surjectivity
A linear transformation is injective if distinct vectors in the domain always have distinct images. For linear maps, injectivity is equivalent to having a trivial kernel.
A transformation is surjective if every vector in the codomain is hit by at least one vector from the domain. For finite-dimensional spaces, surjectivity is closely related to having full rank. When both properties hold, the transformation is an isomorphism between the vector spaces.
4 Operations on linear transformations
Linear transformations themselves can be combined in ways that mirror arithmetic and function composition. These operations preserve linearity and create new linear transformations from old ones.
Such operations make the collection of linear transformations into a rich algebraic object, often studied as a vector space or an algebra of maps.
4.1 Addition of transformations
If two linear transformations have the same domain and codomain, they can be added pointwise: \[ (T+S)(v)=T(v)+S(v). \] The result is again linear.
Addition allows transformations to be blended together. It is useful in constructing new maps from known ones and in studying spaces of operators.
4.2 Scalar multiplication
A linear transformation can also be multiplied by a scalar: \[ (cT)(v)=c\,T(v). \] This operation produces another linear transformation and is compatible with addition.
Scalar multiplication scales the effect of the transformation uniformly. It is a basic part of the vector space structure on the set of all linear maps between fixed spaces.
4.3 Composition of transformations
If the codomain of one linear transformation matches the domain of another, the two can be composed. The composition of linear transformations is again linear.
Composition reflects performing one transformation after another. In matrix terms, it corresponds to matrix multiplication, with order mattering because the operations are generally not commutative.
4.4 Identity and zero transformations
The identity transformation maps every vector to itself. It acts as a neutral element under composition.
The zero transformation sends every vector to the zero vector. It is the additive identity among linear transformations. These two maps serve as reference points for understanding more complicated operators.
5 Special types of linear transformations
Many familiar geometric transformations are linear. These examples help visualize the abstract definitions and illustrate how linear maps behave in concrete settings.
Each special type has characteristic effects on vectors, lengths, angles, or orientation, though not all preserve each geometric feature.
5.1 Identity transformation
The identity transformation leaves every vector unchanged. It is linear and serves as the simplest possible example of a transformation.
Although trivial, it plays an important structural role. It is the baseline against which invertibility and composition are measured.
5.2 Projection
A projection sends vectors onto a subspace, often by discarding components perpendicular to that subspace. Repeated application of a projection gives the same result as applying it once.
Projections are used to isolate parts of vectors, simplify calculations, and model best-fit approximations. They are especially important in geometry and numerical methods.
5.3 Reflection
A reflection flips vectors across a line, plane, or higher-dimensional subspace, depending on the setting. Reflections preserve lengths but reverse orientation in many cases.
They are common in geometry because they produce a clear and intuitive change while remaining linear. Reflections also appear as building blocks for more complicated orthogonal transformations.
5.4 Rotation
A rotation turns vectors about a fixed point or axis while preserving lengths and angles. In Euclidean spaces, rotations are linear transformations when the center is at the origin.
Rotations are fundamental in geometry, physics, and graphics. They model changes in orientation without stretching or skewing the space.
5.5 Scaling
A scaling transformation multiplies vectors by a fixed scalar, enlarging or shrinking them uniformly. If the scalar is negative, the transformation also reverses direction.
Scaling is one of the simplest nontrivial linear transformations. It shows how a linear map can change size while preserving direction along each line through the origin.
5.6 Shear transformations
A shear shifts points parallel to a fixed direction, with the amount of shift depending linearly on position. It preserves areas in some cases but changes angles and shapes.
Shears are important in geometry and computer graphics because they create slanted effects. They also demonstrate that linear transformations need not preserve orthogonality or distance.
6 Eigenvalues and eigenvectors
Eigenvalues and eigenvectors describe directions that are preserved, up to scaling, by a linear transformation. They reveal intrinsic features of the map that are often hidden by a particular matrix representation.
These concepts are central to diagonalization, stability analysis, and many computational methods. They provide a way to understand repeated action of a transformation.
6.1 Eigenvectors
An eigenvector is a nonzero vector that is transformed into a scalar multiple of itself. The direction of such a vector remains unchanged, even if its length is altered.
Eigenvectors identify special invariant directions. Their existence can simplify a problem dramatically, especially when a transformation acts differently on distinct directions.
6.2 Eigenvalues
The scalar associated with an eigenvector is called an eigenvalue. It measures the factor by which the eigenvector is stretched, compressed, or reversed.
Eigenvalues summarize how a transformation acts on special directions. They are often computed from the characteristic polynomial of the associated matrix.
6.3 Eigenspaces
The set of all eigenvectors associated with a particular eigenvalue, together with the zero vector, forms an eigenspace. This set is a subspace of the vector space.
Eigenspaces organize eigenvectors by shared scaling behavior. Their dimensions indicate how many independent directions are associated with a given eigenvalue.
6.4 Diagonalization
A linear transformation is diagonalizable if there exists a basis consisting entirely of eigenvectors. In such a basis, the matrix of the transformation becomes diagonal.
Diagonalization simplifies many computations, especially powers of matrices and iterated transformations. When available, it gives one of the most transparent descriptions of a linear operator.
6.4.1 Conditions for diagonalizability
A transformation is diagonalizable when it has enough linearly independent eigenvectors to form a basis. This depends on the structure and multiplicity of its eigenvalues and eigenspaces.
Not every linear transformation is diagonalizable. When the eigenvectors do not span the space, a more general canonical form may be needed.
6.4.2 Diagonal matrix representation
In a diagonal matrix representation, the action of the transformation on each basis vector is simply multiplication by the corresponding diagonal entry. The off-diagonal entries vanish because the basis vectors do not mix under the map.
This representation is especially convenient because it separates the transformation into independent one-dimensional actions. Many calculations become straightforward in this form.
7 Applications
Linear transformations are used throughout mathematics, science, and engineering because they provide a unified language for describing change. Their usefulness comes from the combination of conceptual clarity and computational tractability.
They appear whenever quantities are related by linear rules or approximated by linear models.
7.1 Solving linear systems
A system of linear equations can be written as a matrix equation involving a linear transformation. Solving the system then becomes equivalent to analyzing whether a given vector lies in the image of the transformation.
The kernel, rank, and invertibility of the associated map reveal the structure of the solution set. This viewpoint is one of the main reasons linear algebra is so effective in applied mathematics.
7.2 Geometry and computer graphics
Linear transformations describe geometric operations on shapes and coordinate systems. Rotations, reflections, scalings, and shears are frequently combined to model motion and deformation.
In computer graphics, these transformations are used to position objects, animate scenes, and convert between coordinate frames. Matrix methods make these tasks efficient and systematic.
7.3 Differential equations
Linear transformations arise in the study of differential equations through linear systems and operators. The behavior of a system can often be understood by examining eigenvalues and eigenvectors of the associated transformation.
This perspective helps characterize growth, decay, oscillation, and stability. Linearization methods also use linear transformations to approximate more complicated dynamics near equilibrium points.
7.4 Data analysis and engineering applications
In data analysis, linear transformations are used for dimensionality reduction, feature extraction, and coordinate changes. In engineering, they model responses of systems, signal processing steps, and mechanical or electrical relations under linear assumptions.
Because linear methods are often easier to compute and interpret than nonlinear ones, they serve as standard tools for approximation and design. Their broad applicability makes them a foundation of modern quantitative analysis.
8 Related concepts
Linear transformations are closely connected to several broader ideas in algebra and geometry. These related notions extend, generalize, or adapt the concept of linearity to other settings.
8.1 Linear operators
A linear operator is a linear transformation from a vector space to itself. Such maps are especially important because their matrices, eigenvalues, and repeated compositions can be studied within one fixed space.
Operators appear in many areas of mathematics, including differential equations and functional analysis. They are often the main objects of interest when the domain and codomain are the same.
8.2 Bilinear maps
A bilinear map is a function that is linear in each argument separately. Unlike a linear transformation, which has one vector input, a bilinear map takes two vector inputs and exhibits linearity in each slot.
Examples include dot products and matrix multiplication viewed in an appropriate abstract form. Bilinear maps generalize the idea of linear dependence to two-variable settings.
8.3 Affine transformations
An affine transformation combines a linear transformation with a translation. Unlike a purely linear map, it need not send the zero vector to itself.
Affine transformations are important in geometry because they model motions and deformations that include shifts of position. They preserve straight lines and parallelism, while relaxing the requirement of fixing the origin.
8.4 Multilinear algebra
Multilinear algebra studies functions that are linear in several variables at once, as well as the tensors and related structures built from them. It extends linear algebra beyond single-input maps.
This field provides the language for advanced topics in geometry, physics, and representation theory. It generalizes the principles of linear transformations to richer algebraic frameworks.