An Inquiry-Based Introduction to Linear Algebra and Applications

Subsection Application: Computer Graphics

As we will discuss, left multiplication by an \(m \times n\) matrix defines a function from \(\R^n\) to \(\R^m\text{.}\) Such a function defined by matrix multiplication is called a matrix transformation. In this section we study some of the properties of matrix transformations and understand how, using the pivots of the matrix, to determine when the output of a matrix transformation covers the whole space \(R^m\) or when a transformation maps distinct vectors to distinct outputs.

Matrix transformations are used extensively in computer graphics to produce animations as seen in video games and movies. For example, consider the dancing figure at left in Figure 7.1. We can identify certain control points (e.g., the point at the neck, where the arms join the torso, etc.) to mark the locations of important points. Using just the control points we can reconstruct the figure. Each control point can be represented as a vector, and so we can manipulate the figure by manipulating the control points with matrix transformations. We will explore this idea in more detail later in this section.

Subsection Introduction

In this section we will consider special functions which take vectors as inputs and produce vectors as outputs. We will use matrix multiplication to produce the output vectors.

If \(A\) is an \(m \times n\) matrix and \(\vx\) is a vector in \(\R^n\text{,}\) then the matrix-vector product \(A \vx\) is a vector in \(\R^m\text{.}\) (Pick some specific \(n, m\) values to understand this statement better.) Therefore, left multiplication by the matrix \(A\) takes an input vector \(\vx\) in \(\R^n\) and produces an output vector \(A\vx\) in \(\R^m\text{,}\) which we will refer to as the image of \(\vx\) under the transformation. This defines a function \(T\) from \(\R^n\) to \(\R^m\) where

These functions are the matrix transformations.

Many of the transformations we consider in this section are from \(\R^2\) to \(\R^2\) so that we can visualize the transformations. As an example, let us consider the transformation \(T\) defined by

If we plot the input vectors \(\vu_1 = \left[ \begin{array}{c} 1 \\ 0 \end{array} \right]\text{,}\) \(\vu_2 = \left[ \begin{array}{c} 0\\1 \end{array} \right]\text{,}\) \(\vu_3 = \left[ \begin{array}{c} 1 \\ 2 \end{array} \right]\text{,}\) and \(\vu_4 = \left[ \begin{array}{r} -1 \\ 1 \end{array} \right]\) (as (blue) circles) and their images \(T(\vu_1) = \left[ \begin{array}{cr} 1\amp 0 \\ 0\amp -1 \end{array} \right]\left[ \begin{array}{c} 1 \\ 0 \end{array} \right] = \left[ \begin{array}{c} 1 \\ 0 \end{array} \right]\text{,}\) \(T(\vu_2) = \left[ \begin{array}{r} 0 \\ -1 \end{array} \right]\text{,}\) \(T(\vu_3) = \left[ \begin{array}{r} 1 \\ -2 \end{array} \right]\text{,}\) and \(T(\vu_4) = \left[ \begin{array}{r} -1 \\ -1 \end{array} \right]\) (as (red) ×'s) on the same set of axes as shown in Figure 7.3, we see that this transformation reflects the input vectors across the \(x\)-axis. We can also see this algebraically since the reflection of the point \((x_1,x_2)\) around the \(x\)-axis is the point \((x_1,-x_2)\text{,}\) and

Subsection Properties of Matrix Transformations

A matrix transformation is a function. When dealing with functions in previous mathematics courses we have used the terms domain and range with our functions. Recall that the domain of a function is the set of all allowable inputs into the function and the range of a function is the set of all outputs of the function. We do the same with transformations. If \(T\) is the matrix transformation \(T(\vx) = A \vx\) for some \(m \times n\) matrix \(A\text{,}\) then \(T\) maps vectors from \(\R^n\) into \(\R^m\text{.}\) So \(\R^n\) is the domain of \(T\) — the set of all input vectors. However, the set \(\R^m\) is only the target set for \(T\) and not necessarily the range of \(T\text{.}\) We call \(\R^m\) the codomain of \(T\text{,}\) while the range of \(T\) is the set of all output vectors. The range of a transformation will come up later, so we make a formal definition.

The range is always a subset of the codomain, but the two sets do not have to be equal. In addition, if a vector \(\vb\) in \(\R^m\) satisfies \(\vb = T(\vx)\) for some \(\vx\) in \(\R^n\text{,}\) then we say that \(\vb\) is the image of \(\vx\) under the transformation \(T\text{.}\) For this reason we also use the term image to also mean the range of a transformation, with the corresponding notation \(\Image(T)\text{.}\)

Because of the properties of the matrix-vector product, if the matrix transformation \(T\) is defined by \(T(\vx) = A\vx\) for some \(m \times n\) matrix \(A\text{,}\) then

and

for any vectors \(\vu\) and \(\vv\) in \(\R^n\) and any scalar \(c\text{.}\) So every matrix transformation \(T\) satisfies the following two important properties:

1. \(T(\vu + \vv) = T(\vu) + T(\vv)\) and

2. \(T(c\vu) = c T(\vv)\text{.}\)

The first property says that a matrix transformation \(T\) preserves sums of vectors and the second that \(T\) preserves scalar multiples of vectors.

As we saw in Activity 7.2, we can combine the two properties of a matrix transformation \(T\) into one: for any scalars \(a\) and \(b\) and any vectors \(\vu\) and \(\vv\) in the domain of \(T\) we have

We can then extend equation (7.1) (by mathematical induction) to any finite linear combination of vectors. That is, if \(\vv_1\text{,}\) \(\vv_2\text{,}\) \(\ldots\text{,}\) \(\vv_k\) are any vectors in the domain of a matrix transformation \(T\) and if \(x_1\text{,}\) \(x_2\text{,}\) \(\ldots\text{,}\) \(x_k\) are any scalars, then

In other words, a matrix transformation preserves linear combinations. For this reason matrix transformations are examples of a larger set of transformation that are called linear transformations. We will discuss general linear transformations in a later section.

There is one other important property of a matrix transformation for us to consider. The functions we encountered in earlier mathematics courses, e.g., \(f(x) = 2x+1\text{,}\) could send the input 0 to any output. However, as a consequence of the definition, any matrix transformation \(T\) maps the zero vector to the zero vector because

Note that the two vectors \(\vzero\) in the last equation may not be the same vector — if \(T : \R^n \to \R^m\text{,}\) then the first \(\vzero\) is in \(\R^n\) and the second in \(\R^m\text{.}\) It should be clear from the context which vector \(\vzero\) is meant.

Subsection Onto and One-to-One Transformations

The problems we have been asking about solutions to systems of linear equations can be rephrased in terms of matrix transformations. The question about whether a system \(A \vx = \vb\) is consistent for any vector \(\vb\) is also a question about the existence of a vector \(\vx\) so that \(T(\vx) = \vb\text{,}\) where \(T\) is the matrix transformation defined by \(T(\vx) = A \vx\text{.}\)

If \(T\) is a matrix transformation, Activity 7.3 illustrates that the range of a matrix transformation \(T\) may not equal its codomain. In other words, there may be vectors \(\vb\) in the codomain of \(T\) that are not the image of any vector in the domain of \(T\text{.}\) If it is the case for a matrix transformation \(T\) that there is always a vector \(\vx\) in the domain of \(T\) such that \(T(\vx) = \vb\) for any vector \(\vb\) in the codomain of \(T\text{,}\) then \(T\) is given a special name.

So the matrix transformation \(T\) from \(\R^n\) to \(\R^m\) defined by \(T(\vx) = A\vx\) is onto if the equation \(A \vx = \vb\) has a solution for each vector \(\vb\) in \(\R^m\text{.}\) Since the vectors \(A\vx\) are linear combinations of the columns of \(A\text{,}\) \(T\) is onto exactly when the span of the columns of \(A\) is all of \(\R^m\text{.}\) Activity 7.3 shows us that \(T\) is onto if every row of \(A\) contains a pivot.

Another question to ask about matrix transformations is how many vectors there can be that map onto a given output vector.

The uniqueness of a solution to \(A \vx = \vb\) is the same as saying that the matrix transformation \(T\) defined by \(T(\vx) = A\vx\) maps exactly one vector to \(\vb\text{.}\) A matrix transformation \(T\) that has the property that every image vector is an image in exactly one way is also a special type of transformation.

So the matrix transformation \(T\) from \(\R^n\) to \(\R^m\) defined by \(T(\vx) = A\vx\) is one-to-one if the equation \(A \vx = \vb\) has a unique solution whenever \(A \vx = \vb\) is consistent. Since the vectors \(A\vx\) are linear combinations of the columns of \(A\text{,}\) the unique solution requirement indicates that any output vector can be written in exactly one way as a linear combination of the columns of \(A\text{.}\) This implies that the columns of \(A\) are linearly independent. Activity 7.4 indicates that this happens when every column of \(A\) is a pivot column.

To summarize, if \(T\) is a matrix transformation defined by \(T(\vx) = A \vx\text{,}\) then \(T\) is onto if every row of \(A\) contains a pivot, and \(T\) is one-to-one if every column of \(A\) is a pivot column. It is important to note the difference: being one-to-one depends on the rows of \(A\) and being onto depends on the columns of \(A\text{.}\)

Having a matrix transformation from \(\R^n\) to \(\R^m\) can tell us things about \(m\) and \(n\text{.}\) For example, when a matrix transformation from \(\R^n\) to \(\R^m\) is one-to-one, it means that there is a unique input vector for every output vector. Since a matrix transformation preserves the algebraic structure of \(\R^n\text{,}\) this implies that the collection of the images of the vectors in the domain of \(T\) form a copy of \(\R^n\) inside of \(\R^m\text{.}\) If we think of \(T\) as a one-to-one matrix transformation with \(T(\vx) = A \vx\) for some \(m \times n\) matrix, then every column of \(A\) will have to be a pivot column. It follows that if there is a one-to-one matrix transformation from \(\R^n\) to \(\R^m\text{,}\) we must have \(m \geq n\text{.}\) Similarly, if a matrix transformation \(T\) from \(\R^n\) to \(\R^m\) is onto, then for each \(\vb\) in \(\R^m\text{,}\) if we select one vector in the domain of \(T\) whose image is \(\vb\text{,}\) then the collection of these vectors in the domain of \(T\) is a copy of \(\R^m\) inside of \(\R^n\text{.}\) So if there is an onto matrix transformation from \(\R^n\) to \(\R^m\text{,}\) then \(n \geq m\text{.}\) As a consequence, the only way a matrix transformation from \(\R^n\) to \(\R^m\) is both one-to-one and onto is if \(n = m\text{.}\)

We conclude this section by adding new equivalent conditions to Theorem 5.3 and Theorem 6.9 from Section 5 and Section 6.

We will continue to add to these theorems, which will eventually give us many different but equivalent perspectives to look at a linear algebra problem. Please keep these equivalent criteria in mind when considering the best possible approach to a problem.

Subsection Examples

What follows are worked examples that use the concepts from this section.

Example 7.10.

A matrix transformation \(T: \R^2 \to \R^2\) defined by

\begin{equation*} T\left(\left[ \begin{array}{c} x\\y \end{array} \right] \right) = \left[ \begin{array}{c} cx\\y \end{array} \right] \end{equation*}

is a contraction in the \(x\) direction if \(0 \lt c \lt 1\) and a dilation in the \(x\) direction if \(c>1\text{.}\)

(a)

Find a matrix \(A\) such that \(T(\vx) = A\vx\text{.}\)

Solution.

Since

\begin{equation*} \left[ \begin{array}{cc} c\amp 0\\0\amp 1 \end{array} \right] \left[ \begin{array}{c} x\\y \end{array} \right] = \left[ \begin{array}{c} cx\\y \end{array} \right]\text{,} \end{equation*}

the matrix \(A = \left[ \begin{array}{cc} c\amp 0\\0\amp 1 \end{array} \right]\) has the property that \(T(\vx) = A\vx\text{.}\)

(b)

Sketch the square \(S\) with vertices \(\vu_1 = \left[ \begin{array}{c} 0\\0 \end{array} \right]\text{,}\) \(\vu_2 = \left[ \begin{array}{c} 1\\0 \end{array} \right]\text{,}\) \(\vu_3 = \left[ \begin{array}{c} 1\\1 \end{array} \right]\text{,}\) and \(\vu_4 = \left[ \begin{array}{c} 0\\1 \end{array} \right]\text{.}\) Determine and sketch the image of \(S\) under \(T\) if \(c = 2\text{.}\)

Solution.

We can determine the image of \(S\) under \(T\) by calculating what \(T\) does to the vertices of \(S\text{.}\) Notice that

\begin{align*} T(\vu_1) \amp = \left[ \begin{array}{cc} 2\amp 0\\ 0\amp 1 \end{array}\right] \left[ \begin{array}{c} 0\\ 0 \end{array} \right] = \left[ \begin{array}{c} 0\\ 0 \end{array} \right]\\ T(\vu_2) \amp = \left[ \begin{array}{cc} 2\amp 0\\ 0\amp 1 \end{array}\right] \left[ \begin{array}{c} 1\\ 0 \end{array} \right] = \left[ \begin{array}{c} 2\\ 0 \end{array} \right]\\ T(\vu_3) \amp = \left[ \begin{array}{cc} 2\amp 0\\ 0\amp 1 \end{array}\right] \left[ \begin{array}{c} 1\\ 1 \end{array} \right] = \left[ \begin{array}{c} 2\\ 1 \end{array} \right]\\ T(\vu_4) \amp = \left[ \begin{array}{cc} 2\amp 0\\ 0\amp 1 \end{array}\right] \left[ \begin{array}{c} 0\\ 1 \end{array} \right] = \left[ \begin{array}{c} 0\\ 1 \end{array} \right] \end{align*}

Since \(T\) is a linear map, the image of \(S\) under \(T\) is the polygon with vertices \((0,0)\text{,}\) \((1,0)\text{,}\) \((2,1)\text{,}\) and \((0,1)\) as shown in Figure 7.11. From Figure 7.11 we can see that \(T\) stretches the figure in the \(x\) direction only by a factor of 2.

Subsection Summary

In this section we determined how to represent any matrix transformation from \(\R^n\) to \(\R^m\) as a matrix transformation, and what it means for a matrix transformation to be one-to-one and onto.

• A matrix transformation is a function \(T: \R^n \to \R^m\) defined by \(T(\vx) = A\vx\) for some \(m \times n\) matrix \(A\text{.}\)

• A matrix transformation \(T\) from \(\R^n\) to \(\R^m\) satisfies

  \begin{equation*} T(a\vu + b\vv) = aT(\vu) + bT(\vv) \end{equation*}

  for any scalars \(a\) and \(b\) and any vectors \(\vu\) and \(\vv\) in \(\R^n\text{.}\) The fact that \(T\) preserves linear combinations is why we say that \(T\) is a linear transformation.

• An \(m \times n\) matrix \(A\) defines the matrix transformation \(T\) via

  \begin{equation*} T(\vx) = A \vx\text{.} \end{equation*}

  The domain of this transformation is \(\R^n\) because the matrix-vector product \(A \vx\) is only defined if \(\vx\) is an \(n \times 1\) vector.

• If \(A\) is an \(m \times n\) matrix, then the codomain of the matrix transformation \(T\) defined by \(T(\vx) = A \vx\) is \(\R^m\text{.}\) This is because the matrix-vector product \(A \vx\) with \(\vx\) an \(n \times 1\) vector is an \(m \times 1\) vector. The range of \(T\) is the subset of the codomain of \(T\) consisting of all vectors of the form \(T(\vx)\) for vectors \(\vx\) in the domain of \(T\text{.}\)

• A matrix transformation \(T\) from \(\R^n\) to \(\R^m\) is one-to-one if each \(\vb\) in \(\R^m\) is the image of at most one \(\vx\) in \(\R^n\text{.}\) If \(T\) is a matrix transformation represented as \(T(\vx) = A\vx\text{,}\) then \(T\) is one-to-one if each column of \(A\) is a pivot column, or if the columns of \(A\) are linearly independent.

• A matrix transformation \(T\) from \(\R^n\) to \(\R^m\) is onto if each \(\vb\) in \(\R^m\) is the image of at least one \(\vx\) in \(\R^n\text{.}\) If \(T\) is a matrix transformation represented as \(T(\vx) = A\vx\text{,}\) then \(T\) is onto if each row of \(A\) contains a pivot position, or if the span of the columns of \(A\) is all of \(\R^m\text{.}\)

Subsection Project: The Geometry of Matrix Transformations

In this section we will consider certain types of matrix transformations and analyze their geometry. Much more would be needed for real computer graphics, but the essential ideas are contained in our examples. A GeoGebra applet is available at geogebra.org/m/rh4bzxee for you to use to visualize the transformations in this project.

Project Activity 7.5.

We begin with transformations that produce the rotated dancing image in Figure 7.1. Let \(R\) be the matrix transformation from \(\R^2\) to \(\R^2\) defined by

\begin{equation*} R\left(\left[ \begin{array}{c} x \\ y \end{array} \right] \right) = \left[\begin{array}{lr} \cos(\theta) \amp -\sin(\theta) \\ \sin(\theta) \amp \cos(\theta) \end{array} \right]\left[ \begin{array}{c} x \\ y \end{array} \right]\text{.} \end{equation*}

These matrices are the rotation matrices.

(a)

Suppose \(\theta = \frac{\pi}{2}\text{.}\) Then

\begin{equation*} R\left(\left[ \begin{array}{c} x \\ y \end{array} \right] \right) = \left[ \begin{array}{cr} 0\amp -1 \\ 1\amp 0 \end{array} \right] \left[ \begin{array}{c} x \\ y \end{array} \right]\text{.} \end{equation*}

(i)

Find the images of \(\vu_1 = \left[ \begin{array}{c} 1 \\ 0 \end{array} \right]\text{,}\) \(\vu_2 = \left[ \begin{array}{c} \frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} \end{array} \right]\text{,}\) and \(\vu_3 = \left[ \begin{array}{c} 0 \\ 1 \end{array} \right]\) under \(R\text{.}\)

(ii)

Plot the points determined by the vectors from part i. The matrix transformation \(R\) performs a rotation. Based on this small amount of data, what would you say the angle of rotation is for this transformation \(R\text{?}\)

(b)

>

Now let \(R\) be the general matrix transformation defined by the matrix

\begin{equation*} \left[\begin{array}{lr} \cos(\theta) \amp -\sin(\theta) \\ \sin(\theta) \amp \cos(\theta) \end{array} \right]\text{.} \end{equation*}

Follow the steps indicated to show that \(R\) performs a counterclockwise rotation of an angle \(\theta\) around the origin. Let \(P\) be the point defined by the vector \(\left[ \begin{array}{c} x\\y \end{array} \right] = \left[ \begin{array}{c}\cos(\alpha) \\ \sin(\alpha) \end{array} \right]\) and \(Q\) the point defined by the vector \(\left[ \begin{array}{c} w\\z \end{array} \right] = \left[ \begin{array}{c} \cos(\alpha+\theta) \\ \sin(\alpha+\theta) \end{array} \right]\) as illustrated in Figure 7.12.

(i)

Use the angle sum trigonometric identities

\begin{align*} \cos(A+B) \amp = \cos(A) \cos(B) - \sin(A) \sin(B)\\ \sin(A+B) \amp = \cos(A) \sin(B) + \cos(B) \sin(A) \end{align*}

to show that

\begin{align*} w \amp = \cos(\theta)x - \sin(\theta)y\\ z \amp = \sin(\theta)x + \cos(\theta)y\text{.} \end{align*}

(ii)

Now explain why the counterclockwise rotation around the origin by an angle \(\theta\) can be represented by left multiplication by the matrix

\begin{equation*} \left[ \begin{array}{rr} \cos(\theta) \amp -\sin(\theta) \\ \sin(\theta) \amp \cos(\theta) \end{array} \right]\text{.} \end{equation*}

Project Activity 7.5 presented the rotation matrices. Other matrices have different effects.

Project Activity 7.6. Different matrix transformations.

(a)

Let \(S\) be the matrix transformation from \(\R^2\) to \(\R^2\) defined by

\begin{equation*} S\left(\left[ \begin{array}{c} x \\ y \end{array} \right] \right) = \left[ \begin{array}{cr} 1\amp 0.5 \\ 0\amp 1 \end{array} \right]\left[ \begin{array}{c} x \\ y \end{array} \right]\text{.} \end{equation*}

Determine the entries of the output vector \(S\left(\left[ \begin{array}{c} x \\ y \end{array} \right] \right)\) and explain the action of the transformation \(S\) on the dancing figure as illustrated in Figure 7.13. (The transformation \(S\) is called a shear in the \(x\) direction.)

(b)

Let \(C\) be the matrix transformation from \(\R^2\) to \(\R^2\) defined by

\begin{equation*} C\left(\left[ \begin{array}{c} x \\ y \end{array} \right] \right) = \left[ \begin{array}{cr} 0.65\amp 0 \\ 0\amp 0.65 \end{array} \right]\left[ \begin{array}{c} x \\ y \end{array} \right]\text{.} \end{equation*}

Determine the entries of the output vector \(C\left(\left[ \begin{array}{c} x \\ y \end{array} \right] \right)\) and explain the action of the transformation \(C\) on the dancing figure as illustrated in Figure 7.14. (The transformation \(C\) is called a contraction.) How would your response change if each \(0.65\) was changed to \(2\) in the matrix \(C\text{?}\)

So far we have seen specific matrix transformations perform a rotations, shears, and contractions. We can combine these, and other, matrix transformations by composition to change figures in different ways, and to created animations of geometric figures. (As we will see later, combining transformations needs to be done carefully in order to obtain the result we want. For example, if we want to first rotate then translate, in what order should the matrices be applied?)