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Mathematical Expression Editor

We define a linear transformation from into and determine whether a given
transformation is linear.

LTR-0010: Introduction to Linear Transformations

We start by reviewing the definition of a function.

Let and be sets. A function from into , denoted by
assigns to each element of , an element of .

The set is called the domain of , and the set is called the codomain.

If , we say that maps to , and is the image of .

The collection of images of all points of is called the range of . More often, we will
refer to the range as the image of or the image of .

In algebra and calculus you worked with functions whose domain and codomain were
each the set of all real numbers. In linear algebra, we call our functions
transformations. The domain and codomain of a transformation are vector spaces. A
typical transformation will map vectors of into .

In this exercise we will introduce a very special type of transformation by
contrasting the effects of two transformations on vectors of . We will see that some
transformations have “nice” properties, while others do not. Define and as
follows:

Each of these transformations takes a vector in , and transforms it to another vector
in . To see if you understand how these transformations are defined, see if you can
determine what these transformations do to the vector .

Now, let’s take the vector and multiply it by a scalar, say .
.

If we apply and to this product, we will see that “handles it better” than . We
compute:

Observe that multiplying the original vector by , then applying , has the same effect
as applying to the original vector, then multiplying the image by . In other
words,

You should try to verify that this property does not hold for transformation . In
other words,

There is nothing special about the number , and it is not hard to prove that for any
scalar and vector of , satisfies

It turns out that satisfies another important property. For all vectors and of we
have:

We leave it to the reader to illustrate this property with a specific example (see
Practice Problem prob:sum). We will show that satisfies (lin2) in general.

Let and , then

It turns out that fails to satisfy this property. Can you prove that this is
the case? Remember that to prove that a property DOES NOT hold, it
suffices to find a counter-example. See if you can find vectors and such that

In Example ex:lintransfirst we were given the images of two vectors, and , under a linear
transformation . Based on this information, we were able to determine the images
of two additional vectors: and . The reason we were able to determine
and is because and can be written as unique linear combinations of and
.

Can every vector of be written as a linear combination of and ?

YesNo

Is the information provided in Example ex:lintransfirst sufficient to determine the image of every
vector in under ?

YesNo

Suppose is a transformation such that
Determine whether is a linear transformation.

Observe that
If were a linear transformation, then we would have:
But according to the given,
Since we conclude that transformation is not linear.

In Exploration init:lintransintro we introduced a transformation which turned out to be non-linear. It
took some work to show that is not linear. The following theorem would have made
our work easier.

Let be a linear transformation. Then . In other words, linear transformations map
the zero vector to the zero vector.

Recall that
was defined by
We evaluate at :
Since , is not linear.

Linear Transformations Induced by Matrices

Let
Let be a vector in . Then is a vector in . We can use this observation to define a
transformation by . For example,

We say that the transformation in Exploration init:matrixtrans is induced by matrix . Such
transformations are often called matrix transformations. In general, an matrix
induces a transformation from into . We will see that all matrix transformations are
linear.

Let be an matrix. Define by . Then is a linear transformation.

Proof

Let and be vectors in , and let be a scalar. By properties of matrix
multiplication we have:
Therefore is a linear transformation.

Let be a linear transformation induced by

(a)

Find the domain and the codomain of .

(b)

Find and

(c)

Find the images of vectors and .

(d)

Find the image of .

item:exlineartrans2a is a matrix, so for the expression to make sense, has to be a vector.
Thus, the domain of is . The product is a vector. So, the codomain of is
.

Observe that the image of is the first column of , and the image of is
the second column of . If you look at the mechanics of how this happened,
you will realize that this is not a coincidence. We will see this phenomenon
again.

item:exlineartrans2d The image of consists of images of all individual vectors in under . Every vector in
can be written as for some real numbers and . Consider the image of
This shows that the range, or the image, of consists of all linear combinations of the
columns of . In other words, the image of is the span of vectors and . The two
vectors are not scalar multiples of each other, therefore they span a plane in
.

Let

(a)

Find the domain and the codomain of the the transformation induced
by .

(b)

Find and draw the image of .

item:lintrans3a is a matrix. So, the domain of is and the codomain is .

Let be an arbitrary vector of . The image of is given by

This shows that the image of every vector in is a scalar multiple of . This means that
the image of is a line in .

Linear Transformations of Subspaces of

Definition def:lin defines a linear transformation as a map from into . We will now make
this definition more general by allowing the domain and the codomain of the
transformation to be subspaces of and . Eventually, a linear transformation will be
defined as a mapping between vector spaces.

Let and be subspaces of and . A transformation is called a linear transformation
if the following are true for all vectors and in , and scalars .

Let be a subspace of consisting of all vectors in the -plane. Let be a subspace of
consisting of all vectors along the -axis. (Do a quick verification that and are
subspaces of .) Define a transformation by
Show that is a linear transformation, and describe its action
geometrically.

Consider two arbitrary elements and of .

Verification of the fact that is similar, and we omit the details.

We have shown that is a linear transformation. The following diagram will help us
visualize the action of geometrically.

We conclude this section by introducing two simple but important transformations.

The identity transformation on , denoted by , is a transformation that maps each
element of to itself.

In other words,
is a transformation such that

The zero transformation, , maps every element of the domain to the zero
vector.

In other words,
is a transformation such that

The identity transformation is linear.

Proof

Left to the reader. (See Practice Problem prob:idtrans)

The zero transformation is linear.

Proof

Left to the reader. (See Practice Problem prob:zerotrans)

For each matrix below, find the domain and the codomain of the linear
transformation induced by ; find and draw the image of . (Hint: See Example ex:lineartrans3.)