4. Summary
Mappings
Definition.
Domain, Range, Image.
Linear Transformations
Definition
Examples
Range, Null Space
Theorem 4.1.
(a) The range T V  of T is a subspace of W.
(b) T maps O V to O W , i.e., T O   O .
Definition:
Null Space
Theorem 4.2.
The null space of T is a subspace of V.
Examples
Theorem 4.3.
Nullity Plus Rank Theorem
If dimV is finite, so is dimT V  and we have
dim N T   dim T V   dimV
(4.1)
Algegraic Operations on Functions, Composition of Functions
Theorem 4.4.
L V ,W  is a linear space with respect to the operations defined in (4.4).
Definition:
Composition of Functions
Theorem 4.5.
Let U, V, W and X be sets. Consider the functions
T :U V
R : W X
S : V W
The compositions are associative, i.e.,
R  ST    RS  T  RST
Definition
Given a function T : V  V , the integral powers of T is defined inductively by
T0  I
T n  TT n1
for n  1 , n  N
where I is the identity function I  x   x .
Theorem 4.6.
Let U, V and W be linear spaces with the same scalars. If
T :U V
S : V W
are linear transformations, so is the composition ST : U  W .
Theorem 4.7.
Let U, V and W be linear spaces with the same scalars.
Let S , T L V ,W  and c be a scalar.
Then
(a) For any function R with values in V, we have
 S  T  R  SR  TR
and
 c S
R c S R
(b) For any linear transformation R : W  U , we have
R  S  T   RS  RT
and
R  c S  c R S
Invertibility / Mapping / Nullity
Definition: Left and Right Inverses
Example:
Theorem 4.8.
A function T : V  W has at most one left inverse.
Furthermore, a left inverse, if exists, is also a right inverse.
Definition: 1-1 function
Theorem 4.9.
A function T : V  W has a left inverse iff T is 1-1 on V.
Definition: Invertible Function
4.7.1. Theorem 4.10.
Let T : V  W be a linear transformation in L V ,W  . The following statements
are then equivalent.
(a) T is 1-1 on V.
(b) T is invertible and the inverse T 1 : T V   V is linear.
(c)
dim N T   0 , i.e., N T   O .
4.7.2. Theorem 4.11.
Let T : V  W be a linear transformation in L V ,W  .
Let n  dimV be finite.
The following statements are then equivalent.
(a) T is 1-1 on V.
(b) If
v ,
1

, v p  are independent elements in V, then T  v1  ,

, T  v p  are
independent elements in T(V).
(c)
dim T V   dimV .
(d) If
v1,

, vn  is a basis in V, so is T  v1  ,

, T  v p  in T(V).
Linear Transformations With Prescribed Values
Theorem 4.12.
Let
v1,
, vn  be a basis for an n-D linear space V. Let
u1,
, un  be n arbitrary
elements in another linear space W. Then there is an unique linear transformation
T :V  W
such that
T  vk   uk
for k  1,
,n
(4.7)
Thus, for an arbitrary x V , we can write
n
n
x   xk vk
so that
k 1
T  x    xk uk
(4.8)
k 1
Matrix Representations
Basis
Diagonal Form
Linear Spaces Of Matrices
Isomorphism Between Linear Transformations And Matrices
Multiplication Of Matrices
Linear Systems, General Solutions
Gauss-Jordan method
Inverses