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Linear map

A linear map, also known as a linear transformation, is a function between two vector spaces over the same field that preserves vector addition and scalar multiplication. Formally, for vector spaces V and W over a field K, a map T: V \to W is linear if it satisfies T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) for all \mathbf{u}, \mathbf{v} \in V and T(\lambda \mathbf{u}) = \lambda T(\mathbf{u}) for all \mathbf{u} \in V and \lambda \in K. This preservation ensures that linear maps maintain the structure of the vector spaces, including the zero vector mapping to the zero vector and the identity map being linear. Linear maps form the foundation of linear algebra, enabling the study of vector spaces through algebraic tools like matrices. When bases are chosen for the domain and codomain, any linear map can be represented by a matrix, and composition of linear maps corresponds to matrix multiplication, facilitating computations in finite-dimensional spaces. Key properties include injectivity (one-to-one, with trivial kernel), surjectivity (onto, with image spanning the codomain), and isomorphism (bijective linear map preserving dimension). The kernel of a linear map, consisting of vectors mapped to zero, and the image, the subspace of outputs, are central subspaces that determine rank and nullity via the rank-nullity theorem. In applications, linear maps model transformations in , such as rotations, scalings, projections, and , which are essential in and engineering. They also underpin differential equations, , and in , where representations solve systems for optimization and . Beyond finite dimensions, linear maps extend to infinite-dimensional spaces, influencing and .

Fundamentals

Definition

In mathematics, particularly in linear , a linear map, also known as a linear , is a between two spaces that preserves the operations of vector addition and scalar multiplication. Formally, let V and W be spaces over the same field \mathbb{F}. A map T: V \to W is linear if it satisfies the following axioms for all vectors u, v \in V and all scalars \alpha \in \mathbb{F}: T(u + v) = T(u) + T(v), \quad T(\alpha u) = \alpha T(u). This definition ensures that linear maps respect the linear structure of the spaces involved. An equivalent formulation of linearity is that T preserves arbitrary finite linear combinations. That is, for any finite collection of vectors v_1, \dots, v_n \in V and scalars \alpha_1, \dots, \alpha_n \in \mathbb{F}, T\left( \sum_{i=1}^n \alpha_i v_i \right) = \sum_{i=1}^n \alpha_i T(v_i). This property follows directly from the additivity and homogeneity axioms by and is often used to verify in practice. The set of all linear maps from V to W is commonly denoted by \mathrm{Hom}(V, W), emphasizing its role as the space of homomorphisms between the vector spaces V and W. The definition of linear maps applies equally to finite-dimensional and -dimensional vector spaces, without requiring additional assumptions such as or boundedness. In finite dimensions, linear maps are often represented using bases and matrices, while in infinite dimensions, they form the foundation for more advanced structures like Hilbert spaces, though the core axioms remain unchanged. The modern axiomatic definition of linear maps emerged in the early , building on 19th-century developments in theory.

Basic properties

Linear maps, by definition, satisfy two fundamental axioms: additivity, T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) for all \mathbf{u}, \mathbf{v} \in V, and homogeneity, T(\alpha \mathbf{u}) = \alpha T(\mathbf{u}) for all scalars \alpha and \mathbf{u} \in V. These axioms separately ensure the preservation of vector addition and , and together they imply the more general property that linear maps preserve arbitrary finite linear combinations: T\left( \sum_{i=1}^n \alpha_i \mathbf{u}_i \right) = \sum_{i=1}^n \alpha_i T(\mathbf{u}_i ) for scalars \alpha_i and vectors \mathbf{u}_i \in V. This follows by on the number of terms, using additivity to handle sums and homogeneity for each scalar multiple./09%3A_Vector_Spaces/9.06%3A_Linear_Transformations) A direct consequence is the preservation of the zero vector: T(\mathbf{0}_V) = \mathbf{0}_W. To prove this, apply additivity to the zero vector itself: T(\mathbf{0}_V) = T(\mathbf{0}_V + \mathbf{0}_V) = T(\mathbf{0}_V) + T(\mathbf{0}_V). Subtracting T(\mathbf{0}_V) from both sides (or equivalently, adding -T(\mathbf{0}_V)) yields \mathbf{0}_W = T(\mathbf{0}_V), as required in the W./09%3A_Vector_Spaces/9.06%3A_Linear_Transformations) Linearity also implies preservation of additive inverses: T(-\mathbf{u}) = -T(\mathbf{u}) for all \mathbf{u} \in V. This follows from additivity applied to \mathbf{u} and its inverse: T(\mathbf{0}_V) = T(\mathbf{u} + (-\mathbf{u})) = T(\mathbf{u}) + T(-\mathbf{u}). Since T(\mathbf{0}_V) = \mathbf{0}_W, it follows that T(-\mathbf{u}) = -T(\mathbf{u})./09%3A_Vector_Spaces/9.06%3A_Linear_Transformations) Finally, linear maps preserve the subspace structure of the domain in the codomain: if U \subseteq V is a , then T(U) = \{ T(\mathbf{u}) \mid \mathbf{u} \in U \} is a subspace of W. To verify this, note that \mathbf{0}_W = T(\mathbf{0}_V) \in T(U) since \mathbf{0}_V \in U. For closure under addition, if T(\mathbf{u}_1), T(\mathbf{u}_2) \in T(U) with \mathbf{u}_1, \mathbf{u}_2 \in U, then T(\mathbf{u}_1) + T(\mathbf{u}_2) = T(\mathbf{u}_1 + \mathbf{u}_2) and \mathbf{u}_1 + \mathbf{u}_2 \in U by the subspace property of U, so T(\mathbf{u}_1 + \mathbf{u}_2) \in T(U). Similarly, for scalar multiplication, \alpha T(\mathbf{u}) = T(\alpha \mathbf{u}) and \alpha \mathbf{u} \in U for \alpha a scalar and \mathbf{u} \in U, ensuring \alpha T(\mathbf{u}) \in T(U)./09%3A_Vector_Spaces/9.08%3A_The_Kernel_and_Image_of_a_Linear_Map)

Examples

Elementary linear maps

The zero map, also known as the trivial linear map, is defined on any vector spaces V and W by T: V \to W, where T(\mathbf{v}) = \mathbf{0} for all \mathbf{v} \in V. This map satisfies the conditions because T(\mathbf{u} + \mathbf{v}) = \mathbf{0} = \mathbf{0} + \mathbf{0} = T(\mathbf{u}) + T(\mathbf{v}) and T(c\mathbf{v}) = \mathbf{0} = c\mathbf{0} = c T(\mathbf{v}) for any scalar c. The identity map provides another fundamental example, defined on a V to itself by I: V \to V, where I(\mathbf{v}) = \mathbf{v} for all \mathbf{v} \in V. It preserves and directly, as I(\mathbf{u} + \mathbf{v}) = \mathbf{u} + \mathbf{v} = I(\mathbf{u}) + I(\mathbf{v}) and I(c\mathbf{v}) = c\mathbf{v} = c I(\mathbf{v}), making it the simplest invertible linear map. A map, or map, acts on a V to itself by T: V \to V, where T(\mathbf{v}) = \alpha \mathbf{v} for a fixed scalar \alpha and all \mathbf{v} \in V. holds since T(\mathbf{u} + \mathbf{v}) = \alpha (\mathbf{u} + \mathbf{v}) = \alpha \mathbf{u} + \alpha \mathbf{v} = T(\mathbf{u}) + T(\mathbf{v}) and T(c\mathbf{v}) = \alpha (c \mathbf{v}) = c (\alpha \mathbf{v}) = c T(\mathbf{v}); if \alpha = 0, this reduces to the zero map, while \alpha = 1 yields the . Projection maps illustrate in spaces, such as the orthogonal onto the x-axis in \mathbb{R}^2, defined by T: \mathbb{R}^2 \to \mathbb{R}^2, where T(x, y) = (x, 0). This satisfies T((x_1, y_1) + (x_2, y_2)) = (x_1 + x_2, 0) = (x_1, 0) + (x_2, 0) = T(x_1, y_1) + T(x_2, y_2) and T(c(x, y)) = (c x, 0) = c (x, 0) = c T(x, y), effectively collapsing the y-coordinate while preserving the x-component. More generally, projections onto subspaces preserve the defining properties of linear maps in finite-dimensional settings. The differentiation operator serves as an example on the finite-dimensional space of polynomials of degree at most n, denoted P_n(\mathbb{F}) over a \mathbb{F}, defined by D: P_n(\mathbb{F}) \to P_n(\mathbb{F}), where D(p(z)) = p'(z). For instance, if p(z) = a_n z^n + \cdots + a_1 z + a_0, then D(p(z)) = n a_n z^{n-1} + \cdots + a_1, which is linear because D(p + q) = (p + q)' = p' + q' = D(p) + D(q) and D(c p) = (c p)' = c p' = c D(p); note that the image lies in P_{n-1}(\mathbb{F}), but the map remains linear within the space. Inclusion maps arise naturally between s, where for a U of a V, the map \iota: U \to V is defined by \iota(\mathbf{u}) = \mathbf{u} for all \mathbf{u} \in U. This is linear since operations in U inherit those from V, satisfying \iota(\mathbf{u}_1 + \mathbf{u}_2) = \mathbf{u}_1 + \mathbf{u}_2 = \iota(\mathbf{u}_1) + \iota(\mathbf{u}_2) and \iota(c \mathbf{u}) = c \mathbf{u} = c \iota(\mathbf{u}), U isometrically into V.

Linear extensions

A linear extension of a linear map is an extension of that map from a subspace to the entire domain vector space while preserving linearity. Consider a linear map T: U \to W, where U is a of a finite-dimensional V over a field K, and W is another vector space over K. To extend T to a linear map \tilde{T}: V \to W, select a basis \{u_1, \dots, u_k\} for U. Since V is finite-dimensional, this basis can be extended to a basis \{u_1, \dots, u_k, v_{k+1}, \dots, v_n\} for V. Define \tilde{T} on the basis by setting \tilde{T}(u_i) = T(u_i) for i = 1, \dots, k and assigning arbitrary values \tilde{T}(v_j) \in W for j = k+1, \dots, n, then extend linearly to all of V. This construction ensures \tilde{T} agrees with T on U and is linear on V. Such extensions always exist when \dim V < \infty, as the basis extension theorem guarantees the completion of any linearly independent set to a basis of V. However, extensions are not necessarily unique; the freedom lies in the choice of values on the additional basis vectors. Uniqueness holds if U admits a complementary subspace S such that V = U \oplus S, in which case every extension \tilde{T} is determined by a unique linear map from S to W, via \tilde{T}(u + s) = T(u) + \tilde{T}(s) for u \in U, s \in S. For a concrete example, let V = \mathbb{R}^2, U = \operatorname{span}\{e_1\} where e_1 = (1,0), and W = \mathbb{R}. Suppose T: U \to \mathbb{R} is defined by T(a e_1) = 2a. Extend the basis \{e_1\} to \{e_1, e_2\} for \mathbb{R}^2, where e_2 = (0,1). Define \tilde{T}(e_1) = 2 and \tilde{T}(e_2) = c for any c \in \mathbb{R}; then \tilde{T}(x,y) = 2x + c y, which extends T. Different choices of c yield different extensions. When the codomain W is the scalar field K, so T is a linear functional, extensions exist more generally, even for infinite-dimensional V, via the algebraic . This theorem states that if f: U \to K is a linear functional on a subspace U of a vector space V over K, then there exists a linear functional F: V \to K such that F|_U = f. The proof relies on applied to the partially ordered set of subspaces containing U with compatible extensions of f, yielding a maximal extension defined on all of V. In the finite-dimensional case, this reduces to the basis extension method described above, without needing .

Matrix representation

Association with matrices

A linear map T: V \to W between finite-dimensional vector spaces V and W over the same field can be associated with a matrix once ordered bases are selected for the domain and codomain. Let \dim V = n and \dim W = m. Choosing a basis B = \{v_1, \dots, v_n\} for V and C = \{w_1, \dots, w_m\} for W, the matrix A of T relative to these bases is the m \times n matrix whose i-th column consists of the coordinates of T(v_i) with respect to C. Explicitly, T(v_i) = \sum_{j=1}^m A_{ji} w_j for each i = 1, \dots, n, where A_{ji} are the entries of A. This construction yields a matrix of size m \times n, reflecting the dimensions of the codomain and domain. The matrix representation extends to arbitrary vectors via coordinate maps. For any x \in V, if _B denotes the coordinate column vector of x with respect to B, then the coordinate vector of T(x) with respect to C satisfies [T(x)]_C = A _B. This relation holds because linearity of T implies that if x = \sum_{k=1}^n \xi_k v_k, then T(x) = \sum_{k=1}^n \xi_k T(v_k) = \sum_{k=1}^n \xi_k \sum_{j=1}^m A_{jk} w_j = \sum_{j=1}^m \left( \sum_{k=1}^n A_{jk} \xi_k \right) w_j, which is precisely the matrix-vector product in coordinates. Thus, the linear map is encoded by standard matrix operations. For fixed bases B and C, this matrix representation is unique, as the coordinates of each T(v_i) are determined solely by the basis expansion. The linearity of T further ensures that the representation is compatible with vector space operations: composition of linear maps corresponds to , and scalar multiples of maps correspond to scalar multiples of their matrices. This association provides a concrete computational framework for studying linear maps in finite dimensions.

Low-dimensional examples

In one dimension, linear maps from \mathbb{R} to \mathbb{R} are simply multiplication by a scalar a, represented by the $1 \times 1 matrix $$. For example, the map T(x) = 2x has matrix \begin{pmatrix} 2 \end{pmatrix}. To verify, applying this matrix to the standard basis vector e_1 = 1 yields $2 \cdot 1 = 2, matching T(1) = 2. In two dimensions, rotations provide a classic example of linear maps preserving lengths and angles. The counterclockwise rotation by an angle \theta in the plane is represented by the matrix \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}. For \theta = 90^\circ, this becomes \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}. Verification on basis vectors: the standard basis e_1 = (1,0) maps to (0,1), and e_2 = (0,1) maps to (-1,0), confirming the right-angle turn. Shear transformations distort shapes by sliding layers parallel to an axis. A horizontal shear in \mathbb{R}^2 that fixes the x-axis and shifts x-coordinates by the y-value is given by \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}. Applying to basis vectors: e_1 = (1,0) remains (1,0), while e_2 = (0,1) maps to (1,1), illustrating the parallel shift. Scaling maps stretch or compress along axes, represented by diagonal matrices. For instance, doubling in the x-direction and halving in the y-direction uses \begin{pmatrix} 2 & 0 \\ 0 & 1/2 \end{pmatrix}. Verification: e_1 maps to (2,0), and e_2 to (0,1/2), recovering the anisotropic scaling. In three dimensions, projections reduce dimensionality by collapsing onto a subspace. The orthogonal projection onto the xy-plane in \mathbb{R}^3 is \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix}. Applying to basis vectors: e_1 = (1,0,0) and e_2 = (0,1,0) remain unchanged, while e_3 = (0,0,1) maps to (0,0,0), confirming the z-coordinate erasure.

Space of linear maps

Vector space structure

The set of all linear maps from a vector space V to a vector space W over the same field, denoted \operatorname{Hom}(V, W), itself forms a vector space with respect to the operations of pointwise addition and scalar multiplication. For any T, S \in \operatorname{Hom}(V, W) and v \in V, addition is defined by (T + S)(v) = T(v) + S(v). This operation is well-defined because the linearity of T and S ensures that T + S is also linear: for any v_1, v_2 \in V and scalar \alpha, (T + S)(\alpha v_1 + v_2) = T(\alpha v_1 + v_2) + S(\alpha v_1 + v_2) = \alpha T(v_1) + T(v_2) + \alpha S(v_1) + S(v_2) = \alpha (T + S)(v_1) + (T + S)(v_2). Scalar multiplication is defined by (\alpha T)(v) = \alpha \, T(v) for any scalar \alpha and v \in V, which similarly preserves linearity due to the properties of T. The additive identity in \operatorname{Hom}(V, W) is the zero map $0: V \to W, defined by $0(v) = 0_W for all v \in V, where $0_W is the zero vector in W. This zero map is linear, as $0(\alpha v_1 + v_2) = 0_W = \alpha 0_W + 0_W = \alpha \, 0(v_1) + 0(v_2). The additive inverse of T \in \operatorname{Hom}(V, W) is (-1)T, satisfying (T + (-1)T)(v) = T(v) + (-T(v)) = 0_W. All vector space axioms hold in \operatorname{Hom}(V, W) because the operations reduce to those in W, and the linearity of maps ensures closure under addition and scalar multiplication. If V and W are finite-dimensional with \dim V = n and \dim W = m, then \dim \operatorname{Hom}(V, W) = n m. This follows from the fact that choosing bases for V and W identifies \operatorname{Hom}(V, W) with the space of n \times m matrices over the field, which has dimension n m. There is a natural isomorphism of vector spaces \operatorname{Hom}(V, W) \cong V^* \otimes W, where V^* is the dual space of V consisting of all linear functionals on V. This isomorphism maps a pure tensor \phi \otimes w \in V^* \otimes W (with \phi \in V^* and w \in W) to the linear map T: V \to W given by T(v) = \phi(v) w for all v \in V, and it extends linearly to the full tensor product.

Endomorphisms and automorphisms

An endomorphism of a vector space V over a field F is a linear map T: V \to V. The set of all endomorphisms of V, denoted \mathrm{End}(V) or \mathrm{Hom}(V, V), is equipped with a ring structure where addition is defined pointwise, (S + T)(v) = S(v) + T(v) for all v \in V, and multiplication is given by composition, (S \circ T)(v) = S(T(v)). This ring operation is associative for composition, distributive over addition, and has the zero map as the additive identity and the identity map as the multiplicative identity. As a special case of the vector space structure on \mathrm{Hom}(V, W), \mathrm{End}(V) forms an algebra over F when scalar multiplication is incorporated. An automorphism of V is an invertible endomorphism, meaning a bijective linear map \phi: V \to V such that there exists an inverse linear map \phi^{-1}: V \to V with \phi \circ \phi^{-1} = \phi^{-1} \circ \phi = \mathrm{id}_V. The set of all automorphisms of V, denoted \mathrm{Aut}(V) or \mathrm{GL}(V), forms a group under composition, known as the general linear group. This group operation is associative, with the identity map serving as the identity element, and every automorphism having an inverse that is also an automorphism. Examples of automorphisms include the identity map \mathrm{id}_V, which satisfies \mathrm{id}_V(v) = v for all v \in V and is clearly invertible. Not all endomorphisms are automorphisms; for instance, nilpotent endomorphisms provide counterexamples. A nilpotent endomorphism N satisfies N^k = 0 (the zero map) for some positive integer k. One such example is the multiplication-by-x operator on the quotient space \mathbb{F}/(x^n), which has basis \{1, x, x^2, \dots, x^{n-1}\} and is defined by T(p)(x) = x \cdot p(x) \mod x^n; this shifts coefficients and is nilpotent since T^n = 0. Another common nilpotent endomorphism is differentiation on the space of polynomials of degree at most n, where repeated application eventually yields the zero polynomial.

Core subspaces and theorems

Kernel and image

The kernel of a linear map T: V \to W between vector spaces over the same field, denoted \ker T, is defined as the set \{v \in V \mid T(v) = 0_W\}, where $0_W is the zero vector in W; this set is also known as the null space of T. The kernel consists of all vectors in the domain V that are mapped to the zero vector in the codomain W. To verify that \ker T is a subspace of V, first observe that T(0_V) = 0_W by the linearity of T, so the zero vector of V belongs to \ker T. Next, if u, v \in \ker T and \alpha is a scalar, then T(u + v) = T(u) + T(v) = 0_W + 0_W = 0_W and T(\alpha u) = \alpha T(u) = \alpha \cdot 0_W = 0_W, showing that \ker T is closed under vector addition and scalar multiplication. Thus, \ker T is a subspace of V. The image of T, denoted \im T, is defined as the set \{T(v) \mid v \in V\}, which represents all vectors in W that are attainable as outputs of T; this set is also called the range of T. The image captures the "span" of the map in the codomain. To establish that \im T is a subspace of W, note that $0_W = T(0_V) \in \im T. For w_1 = T(v_1), w_2 = T(v_2) \in \im T and scalar \alpha, it follows that w_1 + w_2 = T(v_1) + T(v_2) = T(v_1 + v_2) \in \im T and \alpha w_1 = \alpha T(v_1) = T(\alpha v_1) \in \im T, confirming closure under addition and scalar multiplication. Therefore, \im T is a subspace of W. A fundamental relation between these subspaces is that the linear map T induces an isomorphism V / \ker T \cong \im T, as stated by the first isomorphism theorem for vector spaces. The dimension of \ker T is termed the nullity of T, denoted \nullity(T) = \dim(\ker T). Similarly, the dimension of \im T is the rank of T, denoted \rank(T) = \dim(\im T).

Rank–nullity theorem

The rank–nullity theorem states that if T: V \to W is a linear map between finite-dimensional vector spaces over the same field, then \dim V = \dim(\ker T) + \dim(\operatorname{im} T), where \ker T is the kernel of T and \operatorname{im} T is the image of T. The dimension of the kernel is called the nullity of T, denoted \operatorname{nullity}(T), and the dimension of the image is the rank of T, denoted \operatorname{rank}(T). Thus, the theorem can be expressed as \rank(T) + \operatorname{nullity}(T) = \dim V. To prove the theorem, let \{v_1, \dots, v_k\} be a basis for \ker T, where k = \dim(\ker T). Extend this to a basis \{v_1, \dots, v_k, v_{k+1}, \dots, v_n\} for V, with n = \dim V. The set \{T(v_{k+1}), \dots, T(v_n)\} is linearly independent and spans \operatorname{im} T, so it forms a basis for \operatorname{im} T with n - k elements. Therefore, \dim(\operatorname{im} T) = n - k = \dim V - \dim(\ker T), establishing the relation. This theorem does not hold in general for infinite-dimensional vector spaces without additional structure, such as completeness in , as the proof relies on finite bases and may fail due to cardinality issues. For instance, the differentiation operator on the space of polynomials has kernel of dimension 1 but image of the same infinite dimension as the domain. A key application is determining invertibility: T is invertible if and only if \operatorname{rank}(T) = \dim V, which implies \ker T = \{0\}.

Cokernel and index

Cokernel

In linear algebra, given a linear map T: V \to W between vector spaces over a field, the cokernel of T, denoted \operatorname{coker} T, is defined as the quotient space W / \operatorname{im} T, where \operatorname{im} T is the image of T. This construction captures the "failure" of T to be surjective by identifying elements in the codomain that differ by elements in the image. Categorically, the cokernel satisfies a universal property: for any linear map g: W \to U such that g \circ T = 0 (i.e., g vanishes on \operatorname{im} T), there exists a unique linear map \overline{g}: \operatorname{coker} T \to U making the diagram commute, where the canonical projection \pi: W \to \operatorname{coker} T satisfies g = \overline{g} \circ \pi. This property ensures the cokernel is unique up to isomorphism and characterizes it as the coequalizer of T and the zero map in the category of vector spaces. The cokernel relates to exact sequences as follows: the sequence $0 \to \ker T \to V \xrightarrow{T} W \to \operatorname{coker} T \to 0 is exact, extending the short exact sequence $0 \to \ker T \to V \to \operatorname{im} T \to 0. Here, exactness at W means \operatorname{im} T = \ker \pi, confirming the quotient structure. When V and W are finite-dimensional, the dimension of the cokernel is \dim \operatorname{coker} T = \dim W - \dim \operatorname{im} T = \dim W - \operatorname{rank} T. This follows directly from the properties of quotient spaces and the rank of linear maps. For example, if T is surjective, then \operatorname{im} T = W, so \operatorname{coker} T = \{0\}, the trivial vector space.

Index

In the context of endomorphisms, the index provides a measure of the difference between the "deficiencies" in the domain and codomain induced by the operator. For an endomorphism T: V \to V on a vector space V over a field, the index of T is defined as \index(T) = \dim \ker T - \dim \coker T, where the cokernel \coker T is the quotient space V / \im T. This integer-valued invariant, when finite, captures essential information about the operator's invertibility properties, particularly in infinite-dimensional settings. Equivalently, \index(T) = \nullity(T) - (\dim V - \rank(T)), relating the nullity (dimension of the kernel) directly to the rank (dimension of the image). When V is finite-dimensional, the index vanishes for every endomorphism T. This follows from the rank-nullity theorem, which asserts that \dim V = \dim \ker T + \dim \im T, so \dim \coker T = \dim V - \dim \im T = \dim \ker T, yielding \index(T) = 0. In infinite-dimensional spaces, the index is typically considered for Fredholm endomorphisms, which are bounded linear operators with finite-dimensional kernel and cokernel; the index then serves as a topological invariant distinguishing non-invertible operators from the invertible ones (which have index 0). A key property of the index for Fredholm endomorphisms is its invariance under continuous deformations, meaning that if a path of Fredholm operators connects T_0 and T_1, then \index(T_0) = \index(T_1). For instance, consider the unilateral shift operator S on the Hilbert sequence space \ell^2(\mathbb{N}_0), defined by S(e_n) = e_{n+1} for the orthonormal basis \{e_n\}_{n=0}^\infty. Here, \ker S = \{0\} (dimension 0), while \im S has codimension 1 (spanned by e_1, e_2, \dots, missing e_0), so \dim \coker S = 1 and \index(S) = -1. This example illustrates how the index can be nonzero in infinite dimensions, reflecting the operator's failure to be surjective despite being injective.

Algebraic classifications

Monomorphisms and epimorphisms

In the category of vector spaces over a field, a linear map T: V \to W is a monomorphism if and only if it is injective. This property holds equivalently when the kernel of T is the zero subspace, \ker T = \{0\}. For finite-dimensional spaces, T is a monomorphism if and only if its rank equals the dimension of the domain, \operatorname{rank} T = \dim V. Dually, a linear map T: V \to W is an epimorphism if and only if it is surjective. This is equivalent to the image of T being the entire codomain, \operatorname{im} T = W. In finite dimensions, T is an epimorphism if and only if \operatorname{rank} T = \dim W. When V and W are finite-dimensional with \dim V = \dim W and T: V \to V is an endomorphism, the rank-nullity theorem implies that monomorphisms and epimorphisms coincide, so T is injective if and only if it is surjective. A standard example of a monomorphism is the inclusion map i: U \to V for a subspace U \subseteq V, which embeds U injectively into V with trivial kernel. For an epimorphism, consider the canonical projection \pi: V \to V/U onto the quotient space, which is surjective with full image.

Isomorphisms

In the category of vector spaces over a field, a linear map T: V \to W is an isomorphism if it is a bijective linear transformation, meaning it is both injective and surjective. Equivalently, T is an isomorphism if and only if it is both a monomorphism and an epimorphism, as bijectivity for linear maps between vector spaces coincides with these categorical properties. Such a map preserves the vector space structure completely, establishing a one-to-one correspondence between the elements of V and W while respecting addition and scalar multiplication. A key property of an isomorphism T is that its inverse T^{-1}: W \to V exists and is itself a linear map, ensuring that the correspondence can be reversed without altering the linear structure. For finite-dimensional vector spaces V and W, the existence of an isomorphism between them is equivalent to \dim V = \dim W, providing a dimension-based classification up to isomorphism. The composition of two isomorphisms T_1: V \to W and T_2: W \to U is again an isomorphism T_2 \circ T_1: V \to U, and the set of all isomorphisms from V to W (when they exist) forms a group under composition, with the identity map as the identity element and inverses as described. A concrete example of an isomorphism arises in change of basis: given two bases \mathcal{B} and \mathcal{C} for a vector space V, the change-of-basis map that sends the coordinate vectors with respect to \mathcal{B} to those with respect to \mathcal{C} is a linear isomorphism from V to itself, as it bijectively relates the coordinate representations while preserving linearity. When V and W are finite-dimensional and equipped with bases, any linear map T: V \to W has a matrix representation A with respect to these bases, and T is an isomorphism if and only if \det(A) \neq 0. This condition ensures that A is invertible, mirroring the bijectivity of T.

Basis changes

Change of basis formula

When representing a linear map T: V \to W between finite-dimensional vector spaces via matrices, the specific matrix depends on the chosen bases for V and W. Let \beta = \{v_1, \dots, v_n\} be a basis for V and \gamma = \{w_1, \dots, w_m\} for W, with the matrix of T relative to these bases denoted [T]_\beta^\gamma = A, where the columns of A are the coordinates of T(v_j) in the \gamma-basis./13%3A_Diagonalization/13.02%3A_Change_of_Basis) To change bases, consider a new basis \beta' = \{v'_1, \dots, v'_n\} for V and \gamma' = \{w'_1, \dots, w'_m\} for W. The change-of-basis matrix P (from \beta' to \beta) has columns that are the coordinates of the \beta'-basis vectors expressed in the \beta-basis, so the coordinate vector satisfies _\beta = P _{\beta'}. Similarly, the change-of-basis matrix Q (from \gamma' to \gamma) has columns as the \gamma'-basis vectors in \gamma-coordinates, yielding _\gamma = Q _{\gamma'}. Both P and Q are invertible since the bases are. The matrix of T with respect to the new bases is then [T]_{\beta'}^{\gamma'} = Q^{-1} A P. This transformation formula arises because coordinate representations must preserve the linearity of T: for any vector v \in V with new coordinates x' = _{\beta'}, the old coordinates are P x', so [T(v)]_\gamma = A P x', and converting to new output coordinates gives [T(v)]_{\gamma'} = Q^{-1} A P x', confirming the matrix multiplication by Q^{-1} A P yields the correct new coordinates. This proof holds generally for any bases, as it relies only on the invertible change-of-basis matrices encoding the linear isomorphisms between coordinate spaces./13%3A_Diagonalization/13.02%3A_Change_of_Basis) The kernel and image of T, as subspaces of V and W, are intrinsically independent of basis choice, but their matrix representations (e.g., bases or spanning sets) transform under the corresponding change-of-basis matrices. Specifically, if \ker(T) has a basis whose coordinates in \beta form a matrix K, then in \beta' it becomes P^{-1} K, reflecting how the subspace "transforms accordingly" while preserving dimensions and linear relations. The image transforms similarly via Q in the codomain. For a concrete example, consider the linear map T: \mathbb{R}^2 \to \mathbb{R}^2 given by rotation by $90^\circ counterclockwise, with matrix A = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} relative to the standard basis \beta = \gamma = \{e_1 = (1,0), e_2 = (0,1)\}. Now take new bases \beta' = \gamma' = \{v'_1 = (1,1), v'_2 = (0,1)\}. The change-of-basis matrix P = Q = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}, since [v'_1]_\beta = \begin{pmatrix} 1 \\ 1 \end{pmatrix} and [v'_2]_\beta = \begin{pmatrix} 0 \\ 1 \end{pmatrix}. Then P^{-1} = \begin{pmatrix} 1 & 0 \\ -1 & 1 \end{pmatrix}, and the new matrix is A' = P^{-1} A P = \begin{pmatrix} -1 & -1 \\ 2 & 1 \end{pmatrix}. Verifying on basis vectors: T(v'_1) = T(1,1) = (-1,1) = -v'_1 + 2 v'_2, and T(v'_2) = T(0,1) = (-1,0) = -v'_1 + v'_2, matching the columns of A'./13%3A_Diagonalization/13.02%3A_Change_of_Basis)

Similarity of matrices

Two square matrices A and B over a field are similar if there exists an invertible matrix P such that B = P^{-1} A P. This relation is an equivalence relation, partitioning matrices into similarity classes. Similarity arises in the representation of endomorphisms: two matrices represent the same linear map on a vector space V with respect to different bases if and only if they are similar, where P is the change-of-basis matrix./09:_Change_of_Basis/9.02:_Operators_and_Similarity) Similar matrices share key invariants, including:
  • The trace, as \operatorname{tr}(B) = \operatorname{tr}(P^{-1} A P) = \operatorname{tr}(A).
  • The determinant, since \det(B) = \det(P^{-1} A P) = \det(A).
  • The characteristic polynomial, \det(\lambda I - B) = \det(\lambda I - A), implying the same eigenvalues (with algebraic multiplicities).
  • The rank, as similar matrices have isomorphic images and kernels.
Over an algebraically closed field, every square matrix is similar to a unique Jordan canonical form (up to permutation of blocks), so two matrices are similar if and only if they have the same Jordan form. The Jordan form fully classifies similarity classes by specifying the sizes of Jordan blocks for each eigenvalue. Matrices with different minimal polynomials cannot be similar, since similar matrices share the same minimal polynomial—the monic polynomial of least degree annihilating the matrix. For example, the matrix \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} (minimal polynomial \lambda^2) is not similar to \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} (minimal polynomial \lambda), despite both having characteristic polynomial \lambda^2.

Topological aspects

Continuity in normed spaces

In normed vector spaces X and Y, a linear map T: X \to Y is continuous if and only if it is continuous at the zero vector, since linearity ensures that T(v + h) - T(v) = T(h) for any v \in X and perturbation h \in X, making the continuity uniform across the space. This continuity at zero is equivalent to boundedness: there exists a constant M \geq 0 such that \|T v\|_Y \leq M \|v\|_X for all v \in X. The infimum of such constants M defines the operator norm \|T\| = \sup_{\|v\|_X = 1} \|T v\|_Y, which measures the maximum "stretching" of the map and is itself a norm on the space of bounded linear maps. When X is finite-dimensional, every linear map T: X \to Y is automatically bounded and thus continuous, as it admits a matrix representation with respect to bases, and the equivalence of all norms on finite-dimensional spaces ensures uniform boundedness independent of the choice of norm. In contrast, infinite-dimensional normed spaces admit unbounded linear maps, which are necessarily discontinuous. A standard example is the differentiation operator D on the space of polynomials over [0,1] equipped with the supremum norm \|p\| = \sup_{x \in [0,1]} |p(x)|, where D(p) = p'; for the sequence p_n(x) = x^n, we have \|p_n\| = 1 but \|p_n'\| = n, showing that no uniform bound M exists.

Bounded linear operators

In normed linear spaces, a linear operator T: X \to Y between X and Y is bounded if there exists a constant M \geq 0 such that \|T x\|_Y \leq M \|x\|_X for all x \in X. The collection of all bounded linear operators from X to Y, denoted B(X, Y), forms a vector space under pointwise addition and scalar multiplication. When Y = X, the space B(X) is equipped with the operator norm \|T\| = \sup_{\|x\| \leq 1} \|T x\|, making it a complete with respect to this norm. The set B(X) is closed under addition and scalar multiplication, inheriting the vector space structure, and it is also closed under composition: if S, T \in B(X), then S \circ T \in B(X) with \|S \circ T\| \leq \|S\| \|T\|. This endows B(X) with a multiplicative structure, turning it into a unital with the identity operator I as the unit element. In the setting of , which are complete inner product spaces, bounded linear operators admit a distinguished involution known as the adjoint. For a bounded linear operator T: H \to H on a Hilbert space H, the adjoint T^*: H \to H is the unique bounded linear operator satisfying \langle T u, v \rangle = \langle u, T^* v \rangle for all u, v \in H, where \langle \cdot, \cdot \rangle denotes the inner product. The adjoint operation is antilinear in the sense that (c T)^* = \overline{c} T^* for scalars c, and it satisfies (T^*)^* = T and (S T)^* = T^* S^* for composable operators. An operator T \in B(H) is self-adjoint if T = T^*, which implies that T maps real parts to real parts in suitable bases and preserves the inner product structure. Self-adjoint operators are a special case of , where T T^* = T^* T; every self-adjoint operator is normal, as T = T^* implies T T^* = T^* T. However, the converse does not hold; normal operators include self-adjoint, (where T^* T = I), and others. In finite dimensions over the complex numbers, normal operators are unitarily diagonalizable but not necessarily self-adjoint. Normal operators play a central role in spectral decompositions on . The spectral theory of bounded linear operators provides tools to analyze their behavior via complex analysis. For T \in B(X) on a Banach space X, the spectrum \sigma(T) is the set of complex numbers \lambda \in \mathbb{C} such that T - \lambda I is not invertible in B(X). The complement, the resolvent set \rho(T) = \mathbb{C} \setminus \sigma(T), consists of points where the resolvent operator R(\lambda, T) = (T - \lambda I)^{-1} exists as a bounded linear operator, analytic in \lambda \in \rho(T). The spectrum is nonempty, compact, and bounded by \|\lambda\| \leq \|T\| for \lambda \in \sigma(T). Finite-rank operators, those in B(X, Y) with finite-dimensional range, form an ideal in the algebra of bounded s and are always compact. In Banach spaces, every compact operator can be uniformly approximated by finite-rank operators: for any compact K \in B(X, Y) and \epsilon > 0, there exists a finite-rank F such that \|K - F\| < \epsilon. This approximation property underpins the density of finite-rank operators in the compact operators and facilitates the study of operator ideals.

Applications

In geometry and physics

In , linear maps form the core of , which describe changes in , , and while preserving parallelism and ratios of distances along lines. An in \mathbb{R}^n can be expressed as \mathbf{x}' = A\mathbf{x} + \mathbf{b}, where A is an invertible linear map representing the linear part—such as rotations, scalings, or shears—and \mathbf{b} is a translation vector that shifts the origin. This linear component A ensures that straight lines map to straight lines and parallel lines remain parallel, making essential for modeling geometric figures in computer-aided design and spatial analysis. In , linear maps underpin velocity transformations, particularly in where Lorentz boosts act as linear transformations between inertial frames moving at constant relative velocities. A Lorentz boost along the x-axis, for instance, transforms coordinates via a that mixes space and time components while preserving the Minkowski metric, ensuring the invariance of the . These boosts, derived from the postulates of , replace Galilean transformations in Newtonian mechanics and are crucial for describing particle motion and electromagnetic field propagations. Coordinate changes in rely on the matrix, which serves as the of a nonlinear at a point, capturing how local differentials transform under . For a \mathbf{F}: \mathbb{R}^n \to \mathbb{R}^n, the J = \frac{\partial \mathbf{F}}{\partial \mathbf{x}} is the of first partial derivatives, providing the best linear that approximates the change in variables near the point, essential for computing integrals and analyzing in dynamical systems. This facilitates the between coordinate systems, such as from Cartesian to polar, by determining volume scaling factors. In , linear maps enable efficient by representing transformations like rotations, scalings, and projections as multiplications on coordinates, allowing manipulation of polygonal meshes in rendering pipelines. These operations, often composed into a single , map object coordinates to screen space while preserving structural integrity, as seen in and frameworks for and . For low-dimensional illustrations, such as 2D rotations, these maps rotate points around the origin without altering distances from it. A key example in is the use of rotation matrices, which are orthogonal linear maps with 1, describing the orientation changes of a under torque-free motion while conserving . In physics simulations, these 3x3 matrices parameterize the of or mechanical systems, evolving via Euler's equations to model and without deforming the 's shape.

In abstract algebra and beyond

In representation theory, linear maps define the actions through which groups and algebras act on vector spaces, providing a framework to study symmetries abstractly. A representation of a finite group G on a vector space V over a field F is a group homomorphism \rho: G \to \mathrm{GL}(V), where \mathrm{GL}(V) denotes the general linear group of invertible linear endomorphisms of V, ensuring that each group element corresponds to an invertible linear map that preserves the vector space structure. For Lie algebras, a representation of a Lie algebra \mathfrak{g} on V is a Lie algebra homomorphism \rho: \mathfrak{g} \to \mathfrak{gl}(V), where \mathfrak{gl}(V) is the Lie algebra of all linear endomorphisms of V equipped with the commutator bracket [T, S] = TS - ST; this maps each basis element of \mathfrak{g} to a linear map on V while preserving the Lie bracket. Such representations, as explored in foundational works, decompose complex structures into irreducible components, facilitating the classification of finite-dimensional modules. In , the exemplifies a linear map operating on the graded algebra of forms. On a smooth manifold M, the space \Omega^k(M) of smooth k-forms forms a , and the d: \Omega^k(M) \to \Omega^{k+1}(M) is a linear that locally extends the total , satisfying d^2 = 0 and the Leibniz d(\alpha \wedge \beta) = d\alpha \wedge \beta + (-1)^{\deg \alpha} \alpha \wedge d\beta. This antiderivation structure enables the , where closed forms (kernels of d) modulo exact forms (images of d) capture topological invariants, as originally developed in Cartan's generalization of . The of d ensures compatibility with pullbacks under diffeomorphisms, preserving the algebraic properties across coordinate charts. The construction extends linear maps via its , allowing bilinear operations to linearize into maps on tensor spaces. For vector spaces V and W over a F, and linear maps f: V \to V', g: W \to W', the tensor product induces a unique linear map f \otimes g: V \otimes_F W \to V' \otimes_F W' defined by (f \otimes g)(v \otimes w) = f(v) \otimes g(w) on simple tensors and extended linearly; this follows from the universal property, which states that any \phi: V \times W \to Z factors uniquely through a linear map \tilde{\phi}: V \otimes W \to Z such that \phi(v, w) = \tilde{\phi}(v \otimes w). In , this property underpins the extension of representations and operators to tensor powers, as tensor products of representations yield representations of product groups. In , linear operators serve as core tools for solving partial differential equations (PDEs) by framing them within on Banach or Hilbert spaces. Differential operators, such as the Laplacian \Delta: C^\infty(\Omega) \to C^\infty(\Omega), act as unbounded linear maps on Sobolev spaces H^k(\Omega), where existence and uniqueness of solutions to elliptic PDEs like \Delta u = f rely on like self-adjointness and Fredholm alternatives. theory further employs linear operators to generate evolution solutions for parabolic PDEs, such as the \partial_t u = Au where A is a linear with suitable , ensuring well-posedness in L^2 spaces. These applications highlight how linear maps abstractly model boundary value problems, with linking weak solutions to operator ranges. Categorification elevates linear maps from vector spaces to morphisms in abelian categories, enriching algebraic structures with higher-dimensional data. In an abelian category \mathcal{C}, such as the category of modules over a ring, Hom-spaces \Hom_{\mathcal{C}}(A, B) form abelian groups under composition, mirroring vector spaces of linear maps, and exact sequences correspond to kernels and images. Categorification promotes a linear invariant, like a Grothendieck group K_0(\mathcal{C}) generated by isomorphism classes with relations from short exact sequences, to the full category where linear maps become functors or natural transformations; for instance, the category of finite-dimensional representations categorifies the representation ring. This process, as in Khovanov homology, replaces numerical invariants with categorical ones, where decategorification recovers the original linear data via Euler characteristics.