# Linear Map

A linear map is a function between vector spaces that preserves the two operations which give vector spaces their structure: addition and scalar multiplication. Despite this deceptively simple requirement, linear maps form the backbone of nearly all of modern mathematics. Differential operators, rotations of the plane, projections onto subspaces, quantum observables, and the derivative itself are all linear maps. The theory developed in this article --- the kernel-image decomposition, the rank-nullity theorem, matrix representations, and duality --- provides the language in which these diverse phenomena are understood as instances of a single algebraic pattern.

The power of linearity lies in what it makes possible. A linear map is completely determined by its values on a basis, which means that every linear map between finite-dimensional spaces can be encoded as a matrix. This passage from abstract to concrete --- and back --- is the central theme of linear algebra.

## Why Structure Preservation Matters

[motivation] ### The search for well-behaved transformations Consider a function $f \colon \mathbb{R}^2 \to \mathbb{R}^2$ that transforms the plane. Among all possible such [functions](/page/Function) --- reflections, translations, dilations, polynomials, wild discontinuous maps --- which ones can we hope to understand completely? The answer begins with a structural observation: the functions that respect the algebraic structure of $\mathbb{R}^2$ are precisely the ones amenable to systematic analysis. What does "respect the structure" mean? A vector space has two operations: we can add vectors and scale them. A function $\alpha \colon \mathbb{R}^2 \to \mathbb{R}^2$ that preserves both operations satisfies \begin{align*} \alpha(u + v) &= \alpha(u) + \alpha(v), \\ \alpha(\lambda u) &= \lambda \, \alpha(u), \end{align*} for all vectors $u, v \in \mathbb{R}^2$ and all scalars $\lambda \in \mathbb{R}$. These two conditions are the definition of a linear map. ### What fails without linearity To appreciate why these conditions are so powerful, consider what goes wrong without them. Take the squaring function $g \colon \mathbb{R} \to \mathbb{R}$ defined by $g(x) = x^2$. This map is decidedly nonlinear: $g(2 + 3) = 25$, while $g(2) + g(3) = 13$. As a consequence, $g$ has no well-defined kernel in the algebraic sense. The preimage $g^{-1}(\{0\}) = \{0\}$ is a single point, but there is no analogue of the rank-nullity theorem: knowing $\dim(\ker g)$ tells us nothing about the dimension of the image. More critically, a nonlinear function cannot be represented by a matrix. The map $h \colon \mathbb{R}^2 \to \mathbb{R}^2$ given by $h(x, y) = (x^2, xy)$ sends lines through the origin to curves, not lines. There is no hope of finding a $2 \times 2$ matrix $A$ such that $h(v) = Av$ for all $v$. Without matrix representation, we lose the entire computational apparatus of determinants, eigenvalues, and canonical forms. ### The matrix connection For a linear map $\alpha \colon U \to V$ between finite-dimensional vector spaces, choosing a basis for $U$ and a basis for $V$ produces a matrix $A$ such that $\alpha(u)$ is computed by matrix multiplication. Different choices of bases yield different matrices, but all such matrices are related by a change-of-basis transformation. This means that intrinsic properties of $\alpha$ --- its rank, nullity, determinant (when $U = V$), and eigenvalues --- are invariants that do not depend on the coordinate system. The single definition of linearity thus unifies three perspectives: the geometric (transformations of space), the computational (matrix arithmetic), and the abstract (structure-preserving homomorphisms between algebraic objects). [/motivation]

## The Definition of a Linear Map

Throughout this article, let $F$ denote a field (such as $\mathbb{R}$ or $\mathbb{C}$), and let $U$ and $V$ be vector spaces over $F$.

[definition:Linear Map] A **linear map** (also called a **linear transformation** or **homomorphism of vector spaces**) is a function \begin{align*} \alpha \colon U &\to V \end{align*} satisfying the following two conditions for all $u, u' \in U$ and all $\lambda \in F$: 1. **Additivity**: $\alpha(u + u') = \alpha(u) + \alpha(u')$. 2. **Homogeneity**: $\alpha(\lambda u) = \lambda \, \alpha(u)$. Equivalently, $\alpha$ is linear if and only if $\alpha(\lambda u + \mu u') = \lambda \, \alpha(u) + \mu \, \alpha(u')$ for all $\lambda, \mu \in F$ and all $u, u' \in U$. The [set](/page/Set) of all linear maps from $U$ to $V$ is denoted $\operatorname{Hom}(U, V)$ or $\mathcal{L}(U, V)$, and is itself a vector space over $F$ under pointwise addition and scalar multiplication. [/definition]

An immediate consequence of the definition is that every linear map sends the zero vector to the zero vector: setting $\lambda = 0$ in the homogeneity condition gives $\alpha(\mathbf{0}_U) = \alpha(0 \cdot u) = 0 \cdot \alpha(u) = \mathbf{0}_V$. This provides a quick test for non-linearity: any function $f$ with $f(\mathbf{0}) \neq \mathbf{0}$ cannot be linear.

A second fundamental property is that a linear map is completely determined by its action on a basis. If $\{e_1, \ldots, e_n\}$ is a basis for $U$, then for any $u = \lambda_1 e_1 + \cdots + \lambda_n e_n$ we have

\begin{align*} \alpha(u) = \lambda_1 \, \alpha(e_1) + \cdots + \lambda_n \, \alpha(e_n). \end{align*}

Conversely, given any choice of vectors $v_1, \ldots, v_n \in V$, there exists a unique linear map $\alpha \colon U \to V$ satisfying $\alpha(e_i) = v_i$ for each $i$. This is the basis of the matrix representation, which we develop in a later section.

[example:Rotation] **Rotation of the plane.** Fix an angle $\theta \in \mathbb{R}$ and define $R_\theta \colon \mathbb{R}^2 \to \mathbb{R}^2$ by \begin{align*} R_\theta(x, y) = (x \cos \theta - y \sin \theta, \; x \sin \theta + y \cos \theta). \end{align*} To verify linearity, take $u = (x_1, y_1)$, $u' = (x_2, y_2)$, and $\lambda \in \mathbb{R}$. Then \begin{align*} R_\theta(u + u') &= R_\theta(x_1 + x_2, \; y_1 + y_2) \\ &= \bigl((x_1 + x_2)\cos\theta - (y_1 + y_2)\sin\theta, \; (x_1 + x_2)\sin\theta + (y_1 + y_2)\cos\theta\bigr) \\ &= (x_1 \cos\theta - y_1 \sin\theta, \; x_1 \sin\theta + y_1 \cos\theta) + (x_2 \cos\theta - y_2 \sin\theta, \; x_2 \sin\theta + y_2 \cos\theta) \\ &= R_\theta(u) + R_\theta(u'). \end{align*} Homogeneity follows by a similar computation. Therefore $R_\theta$ is a linear map for every $\theta$. [/example]

[example:Differentiation] **The derivative as a linear map.** Let $P_n(\mathbb{R})$ denote the vector space of real polynomials of degree at most $n$, and define the [differentiation](/page/Derivative) map \begin{align*} D \colon P_n(\mathbb{R}) &\to P_{n-1}(\mathbb{R}), \\ p &\mapsto p'. \end{align*} Linearity of $D$ is precisely the statement that the derivative of a sum is the sum of derivatives, and the derivative of a scalar multiple is the scalar times the derivative --- both standard properties from calculus. The map $D$ is surjective: every polynomial $q$ of degree at most $n - 1$ is the derivative of some polynomial in $P_n(\mathbb{R})$. Its kernel consists of the constant polynomials, so $\ker D = P_0(\mathbb{R})$, which has dimension $1$. [/example]

[example:Projection] **Projection onto a subspace.** Let $V = W \oplus W'$ be a direct sum decomposition. The projection onto $W$ along $W'$ is the map \begin{align*} \pi \colon V &\to V, \\ w + w' &\mapsto w, \end{align*} where $w \in W$ and $w' \in W'$. This map is well-defined because every vector in $V$ has a unique decomposition as a sum of an element of $W$ and an element of $W'$. Linearity follows from the uniqueness of this decomposition: if $v_1 = w_1 + w_1'$ and $v_2 = w_2 + w_2'$, then $v_1 + v_2 = (w_1 + w_2) + (w_1' + w_2')$, so $\pi(v_1 + v_2) = w_1 + w_2 = \pi(v_1) + \pi(v_2)$. A projection satisfies $\pi^2 = \pi$: applying it twice gives the same result as applying it once. [/example]

## Kernel and Image

The two most important subsets associated with a linear map are its kernel and its image. Together, they measure how far $\alpha$ is from being injective and surjective, respectively. Their interplay is the subject of the rank-nullity theorem and the [first isomorphism theorem](/theorems/791).

[definition:Kernel] Let $\alpha \colon U \to V$ be a linear map. The **kernel** (or **null space**) of $\alpha$ is the set \begin{align*} \ker \alpha = \{ u \in U : \alpha(u) = \mathbf{0}_V \}. \end{align*} The **nullity** of $\alpha$ is $n(\alpha) = \dim(\ker \alpha)$. [/definition]

[definition:Image] Let $\alpha \colon U \to V$ be a linear map. The **image** (or **range**) of $\alpha$ is the set \begin{align*} \operatorname{im} \alpha = \{ \alpha(u) : u \in U \} = \{ v \in V : v = \alpha(u) \text{ for some } u \in U \}. \end{align*} The **rank** of $\alpha$ is $r(\alpha) = \dim(\operatorname{im} \alpha)$. [/definition]

Both $\ker \alpha$ and $\operatorname{im} \alpha$ are subspaces --- the kernel is a subspace of $U$, the image a subspace of $V$. Verifying this is a direct application of linearity: if $\alpha(u_1) = \mathbf{0}$ and $\alpha(u_2) = \mathbf{0}$, then $\alpha(\lambda u_1 + \mu u_2) = \lambda \alpha(u_1) + \mu \alpha(u_2) = \mathbf{0}$, so $\ker \alpha$ is closed under linear combinations. The argument for $\operatorname{im} \alpha$ is analogous.