Learn how to verify that a transformation is linear, or prove that a transformation is not linear.

Understand the relationship between linear transformations and matrix transformations.

Recipe: compute the matrix of a linear transformation.

Theorem: linear transformations and matrix transformations.

Notation: the standard coordinate vectors

Vocabulary:linear transformation, standard matrix, identity matrix.

In Section 4.1, we studied the geometry of matrices by regarding them as functions, i.e., by considering the associated matrix transformations. We defined some vocabulary (domain, codomain, range), and asked a number of natural questions about a transformation. For a matrix transformation, these translate into questions about matrices, which we have many tools to answer.

In this section, we make a change in perspective. Suppose that we are given a transformation that we would like to study. If we can prove that our transformation is a matrix transformation, then we can use linear algebra to study it. This raises two important questions:

How can we tell if a transformation is a matrix transformation?

If our transformation is a matrix transformation, how do we find its matrix?

for all vectors in and all scalars Since a matrix transformation satisfies the two defining properties, it is a linear transformation

We will see in the next subsection that the opposite is true: every linear transformation is a matrix transformation; we just haven't computed its matrix yet.

Since we have
by the second defining property. The only vector such that is the zero vector.

Let us suppose for simplicity that Then

In engineering, the second fact is called the superposition principle; it should remind you of the distributive property. For example, for any vectors and any scalars To restate the first fact:

A linear transformation necessarily takes the zero vector to the zero vector.

One can show that, if a transformation is defined by formulas in the coordinates as in the above example, then the transformation is linear if and only if each coordinate is a linear expression in the variables with no constant term.

When deciding whether a transformation is linear, generally the first thing to do is to check whether if not, is automatically not linear. Note however that the non-linear transformations and of the above example do take the zero vector to the zero vector.

Find an example of a transformation that satisfies the first property of linearity but not the second.

Subsection4.3.2The Standard Coordinate Vectors

In the next subsection, we will present the relationship between linear transformations and matrix transformations. Before doing so, we need the following important notation.

Standard coordinate vectors

The standard coordinate vectors in are the vectors

The th entry of is equal to 1, and the other entries are zero.

From now on, for the rest of the book, we will use the symbols to denote the standard coordinate vectors.

There is an ambiguity in this notation: one has to know from context that is meant to have entries. That is, the vectors

may both be denoted depending on whether we are discussing vectors in or in

The standard coordinate vectors in and are pictured below.

These are the vectors of length 1 that point in the positive directions of each of the axes.

Multiplying a matrix by the standard coordinate vectors

If is an matrix with columns then for each

In other words, multiplying a matrix by simply selects its th column.

For example,

Definition

The identity matrix is the matrix whose columns are the standard coordinate vectors in

We suppose for simplicity that is a transformation from to Let be the matrix given in the statement of the theorem. Then

The matrix in the above theorem is called the standard matrix for The columns of are the vectors obtained by evaluating on the standard coordinate vectors in To summarize part of the theorem:

Matrix transformations are the same as linear transformations.

Dictionary

Linear transformations are the same as matrix transformations, which come from matrices. The correspondence can be summarized in the following dictionary.