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:
For example, we saw in this example in Section 4.1 that the matrix transformation
is a counterclockwise rotation of the plane by However, we could have defined in this way:
Given this definition, it is not at all obvious that is a matrix transformation, or what matrix it is associated to.
In this section, we introduce the class of transformations that come from matrices.
A linear transformation is a transformation satisfying
for all vectors in and all scalars
Let be a matrix transformation: for an matrix By this proposition in Section 2.4, we have
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.
Let be a linear transformation. 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.
If is a linear transformation, then we define two associated subspaces:
Here, is called the kernel of and is called the image of
Thus, the image is just another name for the range of but the name image is more commonly used in the context of linear maps. Note that if is a matrix transformation, for an matrix then
The rank of a linear transformation written is the dimension of the image
The nullity of a linear transformation written is the dimension of the kernel
In the next subsection, we will present the relationship between linear transformations and matrix transformations. Before doing so, we need the following important notation.
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.
If is an matrix with columns then for each
In other words, multiplying a matrix by simply selects its th column.
For example,
The identity matrix is the matrix whose columns are the standard coordinate vectors in
We will see in this example below that the identity matrix is the matrix of the identity transformation.
Now we can prove that every linear transformation is a matrix transformation, and we will show how to compute the matrix.
Let be a linear transformation. Let be the matrix
Then is the matrix transformation associated with that is, We write
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.
Linear transformations are the same as matrix transformations, which come from matrices. The correspondence can be summarized in the following dictionary.
We saw in the above example that the matrix for counterclockwise rotation of the plane by an angle of is
Recall from this definition in Section 4.1 that the identity transformation is the transformation defined by for every vector
We computed in this example that the matrix of the identity transform is the identity matrix: for every in
Therefore, for all vectors the product of the identity matrix and a vector is the same vector.
