# Definitions and Basic Properties

Throughout this chapter, the scalar field is either always the set or always the set

DEFINITION 4.1.1 (Linear Transformation)   Let and be vector spaces over A map is called a linear transformation if

We now give a few examples of linear transformations.

EXAMPLE 4.1.2
1. Define by for all Then is a linear transformation as

2. Verify that the maps given below from to are linear transformations. Let
1. Define
2. For any define
3. For a fixed vector define Note that examples and can be obtained by assigning particular values for the vector
3. Define by
Then is a linear transformation with and
4. Let be an real matrix. Define a map by

Then is a linear transformation. That is, every real matrix defines a linear transformation from to
5. Recall that is the set of all polynomials of degree less than or equal to with real coefficients. Define by

for Then is a linear transformation.

PROPOSITION 4.1.3   Let be a linear transformation. Suppose that is the zero vector in and is the zero vector of Then

Proof. Since we have

So, as height6pt width 6pt depth 0pt

From now on, we write for both the zero vector of the domain space and the zero vector of the range space.

DEFINITION 4.1.4 (Zero Transformation)   Let be a vector space and let be the map defined by

Then is a linear transformation. Such a linear transformation is called the zero transformation and is denoted by

DEFINITION 4.1.5 (Identity Transformation)   Let be a vector space and let be the map defined by

Then is a linear transformation. Such a linear transformation is called the Identity transformation and is denoted by

We now prove a result that relates a linear transformation with its value on a basis of the domain space.

THEOREM 4.1.6   Let be a linear transformation and be an ordered basis of Then the linear transformation is a linear combination of the vectors

In other words, is determined by

Proof. Since is a basis of for any there exist scalars such that So, by the definition of a linear transformation

Observe that, given we know the scalars Therefore, to know we just need to know the vectors in

That is, for every is determined by the coordinates of with respect to the ordered basis and the vectors height6pt width 6pt depth 0pt

EXERCISE 4.1.7
1. Let and with
2. Let with
3. Let with
4. Let and with
5. Let with
2. Recall that is the space of all matrices with real entries. Then, which of the following are linear transformations
where is some fixed matrix.
3. Let be a map. Then is a linear transformation if and only if there exists a unique such that for every
4. Let be an real matrix. Consider the linear transformation

Then prove that In general, for prove that
5. Use the ideas of matrices to give examples of linear transformations that satisfy:
1. where
6. Let be a linear transformation such that and Let such that Then prove that the set is linearly independent. In general, if for and then for any vector with prove that the set is linearly independent.
7. Let be a linear transformation, and let with Consider the sets

Show that for every there exists such that
8. Define a map by the complex conjugate of Is linear on
(a) over (b) over
9. Find all functions that satisfy the conditions
1. and
2. for all
That is, fixes the line and sends the point for to its mirror image along the line

THEOREM 4.1.8   Let be a linear transformation. For define the set

Suppose that the map is one-one and onto.
1. Then for each the set is a set consisting of a single element.
2. The map defined by

is a linear transformation.

Proof. Since is onto, for each there exists a vector such that So, the set is non-empty.

Suppose there exist vectors such that But by assumption, is one-one and therefore This completes the proof of Part

We now show that as defined above is a linear transformation. Let Then by Part  there exist unique vectors such that and Or equivalently, and So, for any we have

Thus for any

Hence defined as above, is a linear transformation. height6pt width 6pt depth 0pt

DEFINITION 4.1.9 (Inverse Linear Transformation)   Let be a linear transformation. If the map is one-one and onto, then the map defined by

is called the inverse of the linear transformation

EXAMPLE 4.1.10
1. Define by Then is defined by

Note that

Hence, the identity transformation. Verify that Thus, the map is indeed the inverse of the linear transformation
2. Recall the vector space and the linear transformation defined by

for Then is defined as

for Verify that Hence, conclude that the map is indeed the inverse of the linear transformation

A K Lal 2007-09-12