• The unique map of the form T:\{\vec{0}\} \to \{\vec{0}\} is linear. • A prototypical example that gives linear maps their name is a function {{tmath| f: \mathbb{R} \to \mathbb{R}: x \mapsto cx }}, of which the
graph is a line through the origin. • More generally, any
homothety \mathbf{v} \mapsto c\mathbf{v} centered in the origin of a vector space is a linear map (here is a scalar). • The zero map \mathbf x \mapsto \mathbf 0 between two vector spaces (over the same
field) is linear. • The
identity map on any module is a linear operator. • For real numbers, the map x \mapsto x^2 is not linear. • For real numbers, the map x \mapsto x + 1 is not linear (but is an
affine transformation). • If A is a m \times n
real matrix, then A defines a linear map from \R^n to \R^m by sending a
column vector \mathbf x \in \R^n to the column vector . Conversely, any linear map between
finite-dimensional vector spaces can be represented in this manner; see '''', below. • If f: V \to W is an
isometry between real
normed spaces such that f(0) = 0 then f is a linear map. This result is not necessarily true for complex normed space. •
Differentiation defines a linear map from the space of all differentiable functions to the space of all functions. It also defines a
linear operator on the space of all
smooth functions (a linear operator is a
linear endomorphism, that is, a linear map with the same
domain and
codomain). Indeed, \frac{d}{dx} \left( a f(x) + b g(x) \right) = a \frac{d f(x)}{dx} + b \frac{d g( x)}{dx}. • A definite
integral over some
interval is a linear map from the space of all real-valued integrable functions on to . Indeed, \int_u^v \left(af(x) + bg(x)\right) dx = a\int_u^v f(x) dx + b\int_u^v g(x) dx . • An indefinite
integral (or
antiderivative) with a fixed integration starting point defines a linear map from the space of all real-valued integrable functions on \R to the space of all real-valued, differentiable functions on . Without a fixed starting point, the antiderivative maps to the
quotient space of the differentiable functions by the linear space of constant functions. • If V and W are finite-dimensional vector spaces over a field , of respective dimensions and , then the function that maps linear maps f: V \to W to matrices in the way described in '''' (below) is a linear map, and even a
linear isomorphism. • The
expected value of a
random variable is a linear function of the random variable: for random variables X and Y we have E[X + Y] = E[X] + E[Y] and . The
conditional expectation is as well. But the
variance of a random variable is not linear, because for instance {{tmath|1= \text{Var}(aX)=a^2\text{Var}(X) }}. File:Streckung eines Vektors.gif|The function f:\R^2 \to \R^2 with f(x, y) = (2x, y) is a linear map. This function scales the x component of a vector by the factor . File:Streckung der Summe zweier Vektoren.gif|The function f(x, y) = (2x, y) is additive: It does not matter whether vectors are first added and then mapped or whether they are mapped and finally added: f(\mathbf a + \mathbf b) = f(\mathbf a) + f(\mathbf b) File:Streckung homogenitaet Version 3.gif|The function f(x, y) = (2x, y) is homogeneous: It does not matter whether a vector is first scaled and then mapped or first mapped and then scaled: f(\lambda \mathbf a) = \lambda f(\mathbf a)
Linear endomorphisms and isomorphisms If a linear map is a
bijection then it is called a '
. In the case where , a linear map is called a linear endomorphism. Sometimes the term ' refers to this case, but the term "linear operator" can have different meanings for different conventions.
Linear extensions Often, a linear map is constructed by defining it on a subset of a vector space and then to the
linear span of the domain. Suppose X and Y are vector spaces and f : S \to Y is a
function defined on some subset . Then a
of f to X, if it exists, is a linear map F : X \to Y defined on X that
extends f (meaning that F(s) = f(s) for all ) and takes its values from the codomain of . When the subset S is a vector subspace of X then a (-valued) linear extension of f to all of X is guaranteed to exist if (and only if) f : S \to Y is a linear map. In particular, if f has a linear extension to \operatorname{span} S, then it has a linear extension to all of . The map f : S \to Y can be extended to a linear map F : \operatorname{span} S \to Y if and only if whenever n > 0 is an integer, c_1, \ldots, c_n are scalars, and s_1, \ldots, s_n \in S are vectors such that , then necessarily . If a linear extension of f : S \to Y exists then the linear extension F : \operatorname{span} S \to Y is unique and F\left(c_1 s_1 + \cdots c_n s_n\right) = c_1 f\left(s_1\right) + \cdots + c_n f\left(s_n\right) holds for all , and s_1, \ldots, s_n as above. If S is linearly independent then every function f : S \to Y into any vector space has a linear extension to a (linear) map \operatorname{span} S \to Y (the converse is also true). For example, if X = \R^2 and Y = \R then the assignment (1, 0) \to -1 and (0, 1) \to 2 can be linearly extended from the linearly independent set of vectors S := \{(1,0), (0, 1)\} to a linear map on {{tmath|1= \operatorname{span}\{(1,0), (0, 1)\} = \R^2 }}. The unique linear extension F : \R^2 \to \R is the map that sends (x, y) = x (1, 0) + y (0, 1) \in \R^2 to F(x, y) = x (-1) + y (2) = - x + 2 y. Every (scalar-valued)
linear functional f defined on a
vector subspace of a real or complex vector space X has a linear extension to all of . Indeed, the
Hahn–Banach dominated extension theorem even guarantees that when this linear functional f is dominated by some given
seminorm p : X \to \R (meaning that |f(m)| \leq p(m) holds for all m in the domain of ) then there exists a linear extension to X that is also dominated by . == Matrices ==