Page 243 - Matrix Analysis & Applied Linear Algebra

238 Chapter 4 Vector Spaces
4.7 LINEAR TRANSFORMATIONS

The connection between linear functions and matrices is at the heart of our sub-
ject. As explained on p. 93, matrix algebra grew out of Cayley’s observation that
the composition of two linear functions can be represented by the multiplication
of two matrices. It’s now time to look deeper into such matters and to formalize
the connections between matrices, vector spaces, and linear functions defined on
vector spaces. This is the point at which linear algebra, as the study of linear
functions on vector spaces, begins in earnest.

Linear Transformations

Let U and V be vector spaces over a field F ( or C for us).
• A linear transformation from U into V is defined to be a linear
function T mapping U into V. That is,

T(x + y)= T(x)+ T(y) and T(αx)= αT(x) (4.7.1)
or, equivalently,
T(αx + y)= αT(x)+ T(y) for all x, y ∈U,α ∈F. (4.7.2)

• A linear operator on U is defined to be a linear transformation
from U into itself—i.e., a linear function mapping U back into U.

Example 4.7.1

• The function 0(x)= 0 that maps all vectors in a space U to the zero
vector in another space V is a linear transformation from U into V, and,
not surprisingly, it is called the zero transformation.
• The function I(x)= x that maps every vector from a space U back to itself
is a linear operator on U. I is called the identity operator on U.
m×n n×1
• For A ∈ and x ∈ , the function T(x)= Ax is a linear
n m
transformation from into because matrix multiplication satisfies
n
A(αx + y)= αAx + Ay. T is a linear operator on if A is n × n.
• If W is the vector space of all functions from to , and if V is the space
of all differentiable functions from to , then the mapping D(f)= df/dx
is a linear transformation from V into W because
d(αf + g) df dg
= α + .
dx dx dx
• If V is the space of all continuous functions from into , then the
& x
mapping defined by T(f)= f(t)dt is a linear operator on V because
0
x x x
' ' '
[αf(t)+ g(t)] dt = α f(t)dt + g(t)dt.
0 0 0