Page 52 - Linear Algebra Done Right

Chapter 3. Linear Maps
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A linear map from V to W is a function T : V → W with the following
Some mathematicians Definitions and Examples
use the term linear properties:
transformation, which
additivity
means the same as
T(u + v) = Tu + Tv for all u, v ∈ V;
linear map.
homogeneity
T(av) = a(Tv) for all a ∈ F and all v ∈ V.
Note that for linear maps we often use the notation Tv as well as the
more standard functional notation T(v).
The set of all linear maps from V to W is denoted L(V, W). Let’s
look at some examples of linear maps. Make sure you verify that each
of the functions defined below is indeed a linear map:
zero
In addition to its other uses, we let the symbol 0 denote the func-
tion that takes each element of some vector space to the additive
identity of another vector space. To be specific, 0 ∈L(V, W) is
defined by
0v = 0.
Note that the 0 on the left side of the equation above is a function
from V to W, whereas the 0 on the right side is the additive iden-
tity in W. As usual, the context should allow you to distinguish
between the many uses of the symbol 0.
identity
The identity map, denoted I, is the function on some vector space
that takes each element to itself. To be specific, I ∈L(V, V) is
defined by
Iv = v.

differentiation
Define T ∈L(P(R), P(R)) by

Tp = p .

The assertion that this function is a linear map is another way of
stating a basic result about differentiation: (f +g) = f +g and

(af) = af whenever f, g are differentiable and a is a constant.