Section 4.3
Properties of
Linear Transformations
from Rn to Rm
ONE-TO-ONE
TRANSFORMATIONS
A linear transformation T: Rn → Rm is said to be
one-to-one if T maps distinct vectors (points) in
Rn to distinct vectors (points) in Rm.
THREE EQUIVALENT
STATEMENTS
Theorem 4.3.1: If A is an n×n matrix and
TA: Rn → Rn is multiplication by A, then the
following statements are equivalent.
(a)
A is invertible.
(b)
The range of TA is Rn.
(c)
TA is one-to-one.
NOTE: This extends our “big theorem” from
Chapter 2. See Theorem 4.3.4 on page 206.
INVERSE OF A 1-1 LINEAR
OPERATOR
If TA: Rn → Rn is a one-to-one linear operator, then
[T] = A is invertible.
The linear operator TA1 : R  R
inverse of TA.
n
n
is called the
Notation: The inverse of T is denoted by T−1 and
its standard matrix is [T−1] = [T]−1
PROPERTIES OF LINEAR
TRANSFORMATIONS
Theorem 4.3.2: A transformation T: Rn → Rm is
linear if and only if the following relationships
hold for all vectors u and v in Rn and every
scalar c.
(a) T(u + v) = T(u) + T(v)
(b) T(cu) = c T(u)
STANDARD BASIS VECTORS IN
Rn
The standard basis vectors in Rn are given by
e1  (1, 0, 0,  , 0)
e 2  (0, 1, 0,, 0)

e n  (0, 0, 0,, 1)
A THEOREM
Theorem 4.3.3: If T: Rn → Rm is a linear
transformation, and e1, e2, . . . , en are the
standard basis vectors for Rn, then the standard
matrix for T is
[T ]  [T (e1 ) |T (e2 ) ||T (en ) ]
EIGENVALUES AND EIGENVECTORS
OF A LINEAR OPERATOR
If T: Rn → Rn is a linear operator, then a scalar λ
is called an eigenvalue of T if there is a nonzero
x in Rn such that
T(x) = λx
Those nonzero vectors x that satisfy this
equation are called the eigenvectors of T
corresponding to λ.