Page 53 - Linear Algebra Done Right

Definitions and Examples
integration
Define T ∈L(P(R), R) by
1 39
Tp = p(x) dx.
0
The assertion that this function is linear is another way of stating
a basic result about integration: the integral of the sum of two
functions equals the sum of the integrals, and the integral of a
constant times a function equals the constant times the integral
of the function.

multiplication by x 2
Define T ∈L(P(R), P(R)) by Though linear maps are
pervasive throughout
2
(Tp)(x) = x p(x) mathematics, they are
not as ubiquitous as
for x ∈ R.
imagined by some
backward shift confused students who
Recall that F ∞ denotes the vector space of all sequences of ele- seem to think that cos
ments of F. Define T ∈L(F , F ) by is a linear map from R
∞
∞
to R when they write
T(x 1 ,x 2 ,x 3 ,...) = (x 2 ,x 3 ,...). “identities” such as
cos 2x = 2 cos x and
n
from F to F m cos(x + y) =
2
3
Define T ∈L(R , R ) by cos x + cos y.
T(x, y, z) = (2x − y + 3z, 7x + 5y − 6z).
More generally, let m and n be positive integers, let a j,k ∈ F for
n
m
j = 1,...,m and k = 1,...,n, and define T ∈L(F , F ) by
T(x 1 ,...,x n ) = (a 1,1 x 1 +· · ·+a 1,n x n ,...,a m,1 x 1 +· · ·+a m,n x n ).

n
Later we will see that every linear map from F to F m is of this
form.
Suppose (v 1 ,...,v n ) is a basis of V and T : V → W is linear. If v ∈ V,
then we can write v in the form
v = a 1 v 1 +· · ·+ a n v n .

The linearity of T implies that