Linear map

A linear map is a function between vector spaces over the same field that preserves vector addition and scalar multiplication. It therefore preserves the algebraic structure used to form combinations of vectors. Linear maps are fundamental objects of linear algebra, providing a coordinate-independent description of operations that can be represented by matrices in finite dimensions. The term linear transformation is commonly synonymous, although some conventions reserve it for maps from a space to itself. (math.brown.edu)

Definition and basic properties

Let $V$ and $W$ be vector spaces over a field $F$. A function $T:V\to W$ is linear if, for every $u,v\in V$ and every scalar $a\in F$,

These conditions are called additivity and homogeneity. Equivalently,

for all $a,b\in F$. The field matters: the scalars may, for example, be real numbers or complex numbers, and linearity is always understood relative to the chosen field. (math.brown.edu)

Every linear map satisfies $T(0)=0$ and $T(-v)=-T(v)$. More generally, it preserves every finite linear combination:

Consequently, its action on a basis determines its action everywhere. Conversely, arbitrary images assigned to basis vectors extend uniquely to a linear map. (math.brown.edu)

Examples and geometric interpretation

Multiplication by a fixed matrix $A\in F^{m\times n}$ defines a linear map $T:F^n\to F^m$ by $T(x)=Ax$. The identity map and the zero map are basic examples. In real coordinate spaces, rotations about the origin, reflections across subspaces, scalings, and shears are linear. Such maps need not preserve lengths or angles; these require additional properties beyond linearity. (wiki.math.ntnu.no)

For example,

is a shear, represented in standard coordinates by

The map $P(x,y)=(x,0)$ is an orthogonal projection onto the horizontal axis. It collapses an entire direction rather than being invertible. Both examples satisfy the defining identities directly. (wiki.math.ntnu.no)

Linearity also applies to spaces whose elements are functions. On the real vector space of polynomials, differentiation $D(p)=p'$ is linear because the derivative preserves sums and multiplication by constants. Thus linear maps are not restricted to geometric vectors or numerical arrays. (homepages.ucl.ac.uk)

Matrix representation

Suppose $V$ has an ordered basis $B=(v_1,\ldots,v_n)$, and $W$ has an ordered basis $C=(w_1,\ldots,w_m)$. The matrix representing $T$ has as its $j$-th column the coordinates of $T(v_j)$ in $C$. If this matrix is $A$, then

where brackets denote coordinate column vectors. The matrix has $m$ rows and $n$ columns. (math.mit.edu)

The map itself does not depend on the chosen bases, but its matrix generally does. For an endomorphism $T:V\to V$, changing the basis replaces $A$ by $P^{-1}AP$, with $P$ the corresponding change-of-coordinate matrix. Such matrices describe the same operation in different coordinates. Their determinants, traces, and eigenvalues agree, even though their entries can differ. (math.mit.edu)

Kernel, image, and dimension

The kernel and image of $T$ are

They are linear subspaces of $V$ and $W$, respectively. The kernel records directions annihilated by the map; the image records all attainable outputs. For $T(x)=Ax$, these become the null space and column space of $A$. (homepages.ucl.ac.uk)

The rank of $T$ is the dimension of its image, and its nullity is the dimension of its kernel. If $V$ is finite-dimensional, the rank–nullity theorem states

Its rank equals the matrix rank of any representing matrix. The theorem separates the domain’s dimension into directions lost and independent directions retained. (math.brown.edu)

A linear map is injective exactly when its kernel is $\{0\}$, and surjective exactly when its image is its codomain. A bijective linear map is a vector space isomorphism, and its inverse is linear. Between finite-dimensional spaces of equal dimension, injectivity and surjectivity are equivalent. (homepages.ucl.ac.uk)

Operations on linear maps

Maps $V\to W$ form a vector space under pointwise addition and scalar multiplication. If $T:U\to V$ and $S:V\to W$ are linear, their composition $S\circ T$ is linear. With compatible bases, composition corresponds to matrix multiplication, with the later operation’s matrix on the left. A map $V\to V$ is often called a linear operator; its powers describe repeated application. (math.brown.edu)

Linear equations and affine maps

A system of linear equations $Ax=b$ asks which inputs a linear map sends to $b$. It is solvable precisely when $b$ belongs to the image. If $x_0$ is one solution, every solution is of the form

The kernel therefore controls nonuniqueness, while the image controls existence. (wiki.math.ntnu.no)

An affine map has the form $x\mapsto Ax+b$. It is linear only when $b=0$: a nonzero translation violates preservation of the zero vector. Accordingly, the elementary expression $ax+b$, sometimes called a “linear function,” is strictly linear in the vector-space sense only when its constant term vanishes. (wiki.math.ntnu.no)