Linear Transformations - Algebra - University Mathematics Handbook

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 .