University Mathematics Handbook (2015)
X. Algebra
Chapter 9. Linear Transformations
9.1 Transformations
a. Let 
and 
be two nonempty sets. If, for each 
, there is a unique corresponding element 
, it is called transformation from to and is denoted 
or 
.
is the preimage of 
, and is the image of .
The image of transformations is the set of all images of . It is denoted 
.
b. is one-to-one correspondence transformation (denoted 
) if different elements of have different corresponding images, that is, 
or 
.
c. is onto if 
. In other words, every 
has at least one preimage of .
d. Let 
be a transformation from to , and let 
be subset of . Transformation 
is a restruction of on , if its domain of definition is , in which it is defined exactly like , that is, for every 
, 
.
9.2 Isomorphism
a. Definition: Let 
and 
be two vector spaces over field 
. One-to-one correspondence transformation 
of vector space over vector space is called isomorphism if for every 
and 
, there holds:
.
b. Spaces and are isomorphic if the exists isomorphism from to .
c. If spaces and are isomorphic, then, under this isomorphism, a zero of space is transformed to a zero of space .
d. If is an isomorphism from vector space to , then set of vectors 
is linearly dependent if, and only if, the set of images 
is linearly dependent on 
.
e. If spaces and are isomorphic, then a linearly independent set on transforms under the isomorphism to linearly independent set in . Therefore, one basis transforms to another basis and 
.
f. If is an 
-dimensional vector space over field , then space is isomorphic to 
.
g. Every two -dimensional vector spaces over the same field are isomorphic.
9.3 Linear Transformation
a. The transformation from into 
is called a linear transformation, if for every 
and 
there holds:
1. 
2. 
b. 
. That is, in a linear transformation, the image of zero 
of is (zero) 
of .
c. An identity transformation 
, which transforms every vector in to itself, is a linear transformation from on .
d. A zero transformation 
, which transforms every vector of to zero vector of , is a linear transformation from to .
e. If 
is a basis of and 
are vectors (not necessarily linearly independent) in space , then there exists a unique linear transformation 
from to , such that 
, 
.
9.4 Image and Kernel of Linear Transformation
a. If 
is a basis in vector space and is a linear transformation from to , then the image of transformation is:
b. The Kernel, 
, of linear transformation 
is the set of vectors in the images of which are the zero vector of ,
c. If is a linear transformation, then:
1. 
is a subspace of .
2. is a subspace of .
d. If is a linear transformation and is an -dimensional vector space, then
e. 
is the transformation rank of 
.
9.5 Linear Operator
a. A linear operator on is a linear transformation from to itself.
b. Linear operator 
is non-singular if 
. Otherwise, is a singular operator.
c. Linear operator is one-to-one correspondence if, and only if, is non-singular.
d. If and are linear operators on , then operators 
and 
, defined as 
and 
,respectively, are linear operators on .
e. Linear operator is called invertible if there exists operator 
such that 
.
f. If and are invertible operators, then operator 
is invertible, and 
.
g. If is a linear operator in an -dimensional vector space, the following propositions are equivalent:
1. is one-to-one correspondent.
2. is non-singular.
3. is onto.
4. is invertible.
9.6 Matrix Representation of Linear Operator
a. Let and let 
be a basis in . We represent
, 
The -th order square matrix
is the representative matrix of operator with respect to basis , or that 
is a matrix representation of on .
b. If is a linear operator and 
is a basis of , then, for every vector 
, there holds 
.
c. Let 
and be bases in vector space . If 
is a matrix of transformation from to , then, for every linear operator , there holds
when 
are matrix representations of , with respect to bases and .