Linear Map

Overview

Linear maps express the structural part of matrix calculations. A matrix represents a map after bases have been selected; the map itself can be defined without coordinates. Keeping this distinction visible makes kernels, images, and composition intelligible across spaces of polynomials and functions as well as coordinate spaces.

Prerequisites

Use vector spaces over a field $F$, linear combinations, and bases. Domain and codomain must be declared. A formula that preserves real scaling may fail to preserve complex scaling when the same underlying sets are regarded as complex vector spaces.

Definition and conventions

For $F$-vector spaces $V,W$, a map $T:V\to W$ is linear if

for all $v,w\in V$ and $a,b\in F$. Equivalently it preserves addition and scalar multiplication separately. Its kernel is $\{v:T(v)=0\}$ and its image is $\{T(v):v\in V\}$. Both are subspaces.

An endomorphism has $V=W$. An isomorphism is a bijective linear map; its inverse is then linear. An affine map has the form $v\mapsto T(v)+c$ and is linear precisely when $c=0$.

Intuition

A linear map respects how a vector is assembled from other vectors. Knowing its values on a basis determines its values everywhere: expand the input and apply the same coefficients to the basis images. The kernel describes information erased by the map, while the image describes attainable outputs.

Examples and non-examples

Differentiation $D:\mathbb R[x]_{\le2}\to\mathbb R[x]_{\le1}$ is linear. In the bases $(1,x,x^2)$ and $(1,x)$,

Its kernel consists of constant polynomials and its image is the whole codomain. Thus its rank is two and its nullity is one.

The map $x\mapsto x^2$ on $\mathbb R$ is not linear, since its value at $2$ is four rather than twice its value at $1$. Translation by a nonzero vector fails already at zero. Complex conjugation is real-linear but not complex-linear: conjugating $i\cdot1$ gives $-i$, whereas multiplying the conjugate of $1$ by $i$ gives $i$.

Important properties

Every linear map sends zero to zero. It is injective exactly when its kernel is zero: $T(v)=T(w)$ is equivalent to $v-w$ lying in the kernel. For finite-dimensional $V$,

A proof extends a kernel basis to a domain basis; the images of the added vectors form an image basis. Composition corresponds to matrix multiplication only when the intermediate bases match.

Connections and further reading

Quotienting by the kernel gives $V/\ker T\cong\operatorname{im}T$, a model for broader isomorphism theorems. See Hefferon, Chapter Three, Section II, including Definition and Examples and Range Space and Null Space.