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Lecture 16: Linear Transformations¶

[This lecture is based on Prof. Crovella's CS 132 lecture notes.]

16.1 Matrices as functions¶

So far we've been treating the matrix equation

as simply another way of writing the vector equation

However, we'll now think of the matrix equation in a new way:

We will think of $A$ as a function that is "acting on" the vector $x$ to create a new vector $b$.

For example, consider $A=\left[\begin{array}{ccc}2& 1& 1\\ 3& 1& -1\end{array}\right].$ Then we find:

In other words, if $x=\left[\begin{array}{c}1\\ -4\\ -3\end{array}\right]$ and $b=\left[\begin{array}{c}-5\\ 2\end{array}\right]$, then $A$ transforms $x$ into $b$.

Notice what $A$ has done: it took a vector in $R^3$ and transformed it into a vector in $R^2$.

Question. How does this fact relate to the shape of $A$?

$A$ is $2\times 3$ -- that is, $A\in R^{2\times 3}$.

This gives a new way of thinking about solving $Ax=b$.

When we solve $Ax=b$, we are finding the vector(s) $x$ in $R^3$ that are transformed into $b$ in $R^2$ under the "action" of $A$.

Transformations of different sizes¶

For a different $A$, the mapping might be from $R^1$ to $R^3$:

Question. What would the shape of $A$ be in the above case?

Since $A$ maps from $R^1$ to $R^3$, $A\in R^{3\times 1}$.

That is, $A$ has 3 rows and 1 column.

In another case, $A$ could be a square $2\times 2$ matrix.

Then, it would map from $R^2$ to $R^2$.

We have moved out of the familiar world of functions of one variable: we are now thinking about functions that transform a vector into a vector.

Or, put another way, functions that transform multiple variables into multiple variables.

16.2 Transformations¶

Some terminology:

A transformation (or function or mapping) $T$ from $R^n$ to $R^m$ is a rule that assigns to each vector $x$ in $R^n$ a vector $T\left(x\right)$ in $R^m$.

The set $R^n$ is called the domain of $T$, and $R^m$ is called the codomain of $T$.

The notation:

indicates that the domain of $T$ is $R^n$ and the codomain is $R^m$.

For $x$ in $R^n,$ the vector $T\left(x\right)$ is called the image of $x$ (under $T$). We may sometimes write that $T$ maps $x\mapsto T\left(x\right)$.

The set of all images $T\left(x\right)$ is called the range of $T$.

Here, the green plane is the set of all points that are possible outputs of $T$ for some input $x$.

So in this example:

• The domain of $T$ is $R^2$
• The codomain of $T$ is $R^3$
• The range of $T$ is the green plane.

Visualizing a transformation¶

Let's do an example. Say that I have these points in $R^2$:

These points form a square in 2-dimensional space, where each side is of length 1.

Now let's transform each of these points according to the following rule. Consider the following matrix, which is called a shear matrix.

We define $T\left(x\right)=Ax$. Then we have

What is the image of each of these points under $T$?

This sort of transformation, where points are successively slid sideways, is called a shear transformation.

16.3 Linear Transformations¶

By the algebraic properties of matrix-vector multiplication, we know that the transformation $x\mapsto Ax$ has the properties that

for all $u,v$ in $R^n$ and all scalars $c$.

We are now ready to define one of the most fundamental concepts in the course: the concept of a linear transformation. (This is why the subject is called linear algebra!)

Definition. A transformation $T$ is linear if:

1. $T(u+v)=T\left(u\right)+T\left(v\right)$ for all $u,v$ in the domain of $T$; and
2. $T\left(cu\right)=cT\left(u\right)$ for all scalars $c$ and all $u$ in the domain of $T$.

To fully grasp the significance of what a linear transformation is, don't think of just matrix-vector multiplication. Think of $T$ as a function in more general terms.

The definition above captures a lot of transformations that are not matrix-vector multiplication. For example, think of:

$T$ is a linear transformation!

Properties of Linear Transformations¶

A key aspect of a linear transformation is that it preserves the operations of vector addition and scalar multiplication.

For example: for vectors $u$ and $v$, one can either:

1. Transform them both according to $T\left(\right)$, then add them, or:
2. Add them, and then transform the result according to $T\left(\right)$.

One gets the same result either way. The transformation and the addition "commute" with each other.

This leads to two important facts.

If $T$ is a linear transformation, then

and

In fact, if a transformation satisfies the second equation for all $u,v$ and $c,d,$ then it must be a linear transformation.

Both of the rules defining a linear transformation derive from this single equation.

Example.

Given a scalar $r$, define $T:R^2\to R^2$ by $T\left(x\right)=rx$.

($T$ is called a contraction when $0\le r\le 1$ and a dilation when $r>1$.)

Let $r=3$, and show that $T$ is a linear transformation.

Solution.

Let $u,v$ be in $R^2$ and let $c,d$ be scalars. Then

Thus $T$ is a linear transformation because it satisfies the rule $T(cu+dv)=cT\left(u\right)+dT\left(v\right)$.

Example.

Let $T\left(x\right)=x+b$ for some $b\ne 0$.

What sort of operation does $T$ implement?

Answer: translation.

Is $T$ a linear transformation?

Solution.

We only need to compare

to

So:

and

If $b\ne 0$, then the above two expressions are not equal.

So $T$ is not a linear transformation.

A Non-Geometric Example: Manufacturing¶

A company manufactures two products, B and C. To do so, it requires materials, labor, and overhead.

For one dollar's worth of product B, it spends 45 cents on materials, 25 cents on labor, and 15 cents on overhead.

For one dollar's worth of product C, it spends 40 cents on materials, 30 cents on labor, and 15 cents on overhead.

Let us construct a "unit cost" matrix:

Let $x=\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$ be a production vector, corresponding to $x_1$ dollars of product B and $x_2$ dollars of product C.

Then define $T:R^2\to R^3$ by

The mapping $T$ transforms a list of production quantities into a list of total costs.

The linearity of this mapping is reflected in two ways:

1. If production is increased by a factor of, say, 4, ie, from $x$ to $4x$, then the costs increase by the same factor, from $T\left(x\right)$ to $4T\left(x\right)$.
2. If $x$ and $y$ are production vectors, then the total cost vector associated with combined production of $x+y$ is precisely the sum of the cost vectors $T\left(x\right)$ and $T\left(y\right)$.

16.4 The Matrix of a Linear Transformation¶

We have seen that every matrix multiplication is a linear transformation from vectors to vectors.

But, are there any other possible linear transformations from vectors to vectors?

No! In other words, the reverse statement is also true:

Every linear transformation from vectors to vectors is a matrix multiplication.

We'll now prove this fact. We'll do it constructively, meaning we'll actually show how to find the matrix corresponding to any given linear transformation $T$.

Theorem. Let $T:R^n\to R^m$ be a linear transformation. Then there is (always) a unique matrix $A$ such that:

Let $e_j$ denote the $j$th column of the identity matrix $I$ in $R^n$. These vectors are called the standard basis vectors.

Then, $A$ is the $m\times n$ matrix whose $j$th column is the vector $T\left(e_j\right)$, where

$A$ is called the standard matrix of $T$.

Proof. Write

Because $T$ is linear, we have:

So ... we see that the ideas of matrix multiplication and linear transformation of vector spaces are essentially equivalent.

In this course, we will use the term linear transformation when we want to focus on a property of the mapping, while the term matrix multiplication focuses on how such a mapping is implemented.

Recap¶

To find the standard matrix of a linear tranformation, ask what the transformation does to the columns of $I$.

Now, in $R^2$, $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$. So:

To find the matrix of any given linear transformation of vectors in $R^2$, we only have to know what that transformation does to these two points.

This is a hugely powerful tool. If we start from some given linear transformation, we can use this idea to find the matrix that implements that linear transformation.

16.5 Rotations¶

For example, let's consider rotation about the origin as a kind of transformation.

First things first: Is rotation a linear transformation?

Recall that a for a transformation to be linear, it must be true that $T(u+v)=T\left(u\right)+T\left(v\right).$

I'm going to show you a "geometric proof."

This figure shows that "the rotation of $u+v$ is the sum of the rotation of $u$ and the rotation of $v$".

So the answer is yes, rotation is a linear transformation.

Let's see how to compute the linear transformation that is a rotation.

Specifically: Let $T:R^2\to R^2$ be the transformation that rotates each point in $R^2$ about the origin through an angle $\theta$, with counterclockwise rotation for a positive angle.

Let's find the standard matrix $A$ of this transformation.

Question. The columns of $I$ are $e_1=\left[\begin{array}{c}1\\ 0\end{array}\right]$ and $e_2=\left[\begin{array}{c}0\\ 1\end{array}\right].$ Where does $T$ map the standard basis vectors?

Referring to the diagram below, we can see that $\left[\begin{array}{c}1\\ 0\end{array}\right]$ rotates into $\left[\begin{array}{c}\cos \theta \\ \sin \theta \end{array}\right],$ and $\left[\begin{array}{c}0\\ 1\end{array}\right]$ rotates into $\left[\begin{array}{c}-\sin \theta \\ \cos \theta \end{array}\right].$

So by the Theorem above,

In particular, a 90 degree counterclockwise rotation corresponds to the matrix

Computing the matrix corresponding to a rotation¶

To demonstrate computationally the use of a rotation matrix, let's rotate the following shape note. It is constructed using an array of 26 vectors in $R^2$ that define its outline. In other words, it is a 2 $\times$ 26 matrix.

To rotate note we need to multiply each column of note by the rotation matrix $A$.

Recall that: in Python matrix multiplication is performed using the @ operator. That is, if A and B are matrices, then

A @ B

will multiply A by every column of B, and the resulting vectors will be formed into a matrix.