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 .