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

We now give a few examples of linear transformations.

- Define
by
for all
Then
is a linear transformation as
- Verify that the maps given below from
to
are
linear transformations. Let
- Define
- For any define
- For a fixed vector define Note that examples and can be obtained by assigning particular values for the vector

- Define
by

Then is a linear transformation with and - Let
be an
real matrix. Define a
map
by
- Recall that
is the set of all polynomials of degree less than or equal to
with real coefficients. Define
by

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.

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

*In other words,
is determined by
*

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

- Which of the following are linear transformations
Justify your
answers.
- Let and with
- Let with
- Let with
- Let and with
- Let with

- Recall that
is the space of all
matrices with real entries.
Then, which of the following are linear transformations
- where is some fixed matrix.

- Let be a map. Then is a linear transformation if and only if there exists a unique such that for every
- Let
be an
real matrix.
Consider the linear transformation
- Use the ideas of matrices to give examples of linear
transformations
that satisfy:
- where

- 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.
- Let
be a
linear transformation, and let
with
Consider the sets
- Define a map
by
the complex conjugate of
Is
linear on
- (a) over (b) over

- Find all functions
that satisfy the conditions
- and
- for all

Is this function a linear transformation? Justify your answer.

- Then for each the set is a set consisting of a single element.
- The map
defined by

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

- Define
by
Then
is defined by

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

A K Lal 2007-09-12