Linear transformations - Transformations - Two-Dimensional Calculus

Two-Dimensional Calculus (2011)

Chapter 3. Transformations

14. Linear transformations

Definition 14.1 A linear transformation is a transformation of the form

Image

where a, b, c, d are constants. The transformation is uniquely determined by the array of coefficients

Image

which is called the matrix of the transformation.

The study of linear transformations from an algebraic point of view forms a basic part of “linear algebra” or “matrix theory.”

In this section, we investigate the geometric properties of linear transformations. Although it may seem that linear transformations are quite special, we shall see later that many questions concerning general transformations can be answered by referring to related linear transformations.

The first thing to observe is that solving the simultaneous linear equations

Image

is precisely equivalent to finding those points in the x, y plane which map onto the point (e, f) in the u, υ plane under the transformation (14.1).

We use the following standard notation

Image

This number Δ is called the determinant of the transformation (14.1), as well as of the matrix (14.2) and of the system (14.3). Two cases must be distinguished.

Case 1. Δ ≠ 0. In this case, for each choice of the right-hand side of (14.3) , there is a unique solution, namely

Image

In terms of the transformation (14.1), this means that each point (e, f) in the u, υ plane is the image of one and only one point in the x, y plane, or equivalently, that F maps the x, y plane one-to-one onto the u, υ plane. Equations (14.5) are in fact the equations of the inverse mapping of F, which we may write as

Image

Note that the map F-1 is a linear transformation of the u, υ plane into the x, y plane. Its matrix is

Image

and its determinant is

Image

Case 2.Δ = 0. Within case 2 there are two possibilities:

a. a, b, c, d are all zero. Then the transformation (14.1) simply maps the whole x, y plane onto the origin of the u, υ plane.

b.Not all coefficients are zero. Suppose that either a or c is different from zero. Multiply the first equation of (14.1) by c, the second by a, and subtract. Using Δ = 0, we find

Image

which is the equation of a straight line through the origin. Thus, the image of every point (x, y) lies on this straight line. (In terms of the system of Equations (14.3), a solution exists for a given right-hand side if and only if the point (e, f) lies on this line; that is, if and only if ceaf = 0.) If either b or d is different from zero, we find similarly that the whole plane maps onto the line du = 0.

Definition 14.2 The linear transformation (14.1) (or the matrix (14.2)) is called singular if Δ = 0 and nonsingular if Δ ≠0.

We may summarize the above discussion as follows. Under a singular linear transformation the whole plane maps into a straight line (or in the extreme case when all the coefficients are zero, into a single point). The mapping is neither one-to-one nor onto. A nonsingular linear transformation is both one-to-one and onto. It has an inverse, which is again a nonsingular linear transformation.

Example 14.1

Image

Matrix is

Image

Δ = 2.F is nonsingular.

Example 14.2

Image

Matrix is

Image

Δ = −1. F is nonsingular.

Example 14.3

Image

Matrix

Image

Δ = − 1. F is nonsingular.

Example 14.4

Image

Matrix

Image

Δ = 0. F is singular. For all x, y, we have 3u + 2υ = 0.

Nonsingular Transformations

For the remainder of this section we restrict our attention to the case of nonsingular transformations. The introduction of polar coordinates once again proves advantageous. Let us set

Image

Then Eqs. (14.1) become

Image

Thus

Image

If we hold θ fixed and let r vary, it follows from Eq. (14.10) that φ is constant, which means geometrically that each ray starting at the origin in the x, y plane maps into a ray from the origin in the u, υ plane (Fig. 14.1). We ask the question: “if the ray in the x, y plane is rotated about the origin in the positive direction what happens to the image ray in the u, υ plane?” More precisely, we may ask: “how does the angle φ depend on θ?” The answer is obtained simply by differentiating Eq. (14.10) with respect to θ.

Image

or

Image

It follows that dφ/dθ has the same sign as the determinant Δ. When Δ is positive we have dφ/dθ > 0 and φ is an increasing function of θ, while for Δ negative, φ is a decreasing function of θ. We may summarize the situation geometrically as follows.

Case 1. Δ > 0. As a ray in the x, y plane is rotated in the counterclockwise direction (θ increasing), the image ray in the u, υ plane rotates in the same direction. The transformation is called orientation preserving (Fig. 14.2).

Case 2. Δ < 0. As a ray in the x, y plane rotates in the counterclockwise direction, the image ray in the u, υ plane rotates in the opposite direction. The transformation is called orientation reversing (Fig. 14.3).

Image

FIGURE 14.1 Correspondence between polar angles

Example 14.1a

Since Δ > 0 in Example 14.1 the transformation is orientation preserving.

Example 14.2a

The transformation in Example 14.2 is orientation reversing. In fact, Eq. (14.10) becomes tan φ = − tan θ. It follows that, in this example, /dθ = − 1, so that the image ray rotates at the same rate in the opposite direction as the ray in the x, y plane. The reason is cleargeometrically if we recall from Example 13.7 that this transformation corresponds to a reflection in the horizontal axis.

Image

FIGURE 14.2 Orientation-preversing transformation

Image

FIGURE 14.3 Orientation-reversing transformation

Example 14.3a

The transformation in Example 14.3 is also orientation reversing. Equation (14.10) becomes tan φ = cot θ, and again / = − 1. As in Example 14.2a the reason is clear geometrically if we recall from Example 13.8 that this transformation corresponds to a reflection across the diagonal.

We should remark that actually every straight line through the origin in the x, y plane maps into a straight line through the origin in the u, υ plane. We have simply restricted our discussion to rays, or half-lines, in order to single , out a direction along each line.

We now return to Eqs. (14.9) and ask what happens if we hold r fixed and let θ vary. Geometrically, we are looking for the image of a circle about the origin in the x, y plane. Although this question can be answered directly by examining Eqs. (14.9), it turns out to be easier to use the device mentioned in Sect. 13, and ask what is the inverse image of a circle in the u, υ plane. If we hold R fixed, we want the set of points in the x, y plane satisfying u2 + υ2 = R2. Returning to Eqs. (14.1), we find that x and y must satisfy

Image

or

Image

where

Image

Equation (14.13) defines a curve whose nature depends on the sign of the quantity AC − B2 (see Corollary 3 of Th. 10.2). But form (14.14)

Image

Also A + C = (a2+b2+c2+d2)/ R2> 0. Thus Ax2 + 2Bxy+Cy2 is a positive define quadratic form, and Eq. (14.13) represents on ellipse. Furtheremore, we know from corollary 4 of Th. 10.2 that the area Image inside the ellipse is

Image

so that

Image

We may summarize as follows.

Under a nonsingular linear transformation the inverse image of each circle with center at the origin in the u, υ plane is an ellipse centered at the origin of the x, y plane. The absolute value of the determinant Δ is equal to the ratio of the area inside the circle to the area inside the ellipse (Fig. 14.4). Note in particular that this ratio of the areas does not depend on R ; it is the same for all circles.

Image

FIGURE 14.4Inverse image of circle

We now may return to our original question concerning the image of a circle in the x, y plane. The answer is provided by the simple observation that the inverse mapping G = F−1is also a nonsingular transformation, and the image of the circle x2 + y2 = r2 under F is the same as its inverse image under G (Fig. 14.5). Applying the above reasoning to the linear transformation G, we see that the circle x2 + y2 = r2 corresponds to an ellipse about the origin in the u, υ plane, and that the area Image' inside the ellipse satisfies

Image

where Δ' is the determinant of G. But Eq. (14.8) relates the determinant of the inverse transformation to that of the original. We find therefore, that

Image

Comparing Eqs. (14.16) and (14.17), we see that in both cases the magnitude of Δ represents the ratio of the area in the u, v plane to the area in the x, y plane. This interpretation of the absolute value of Δ together with our earlier discussion of the significance of the sign of Δ, allows us to describe the value of the determinant Δ in a purely geometric fashion. This description is fundamental to the study of differentiable mappings in the same way as the interpretation of the magnitude and sign of the slope of a straight line is to the study of differentiable functions of a single variable.

Image

FIGURE 14.5Image of a circle

Example 14.5

Image

A circle of radius r in the x, y plane may be written in parametric form as x = r cos t, y = r sin t. Its image is then u = r cos t, υ = 2r sin t, which is the parametric form of the ellipse (Fig. 14.6)

Image

The area inside this ellipse is πr.(2r) = 2πr2 which is twice the area inside the original circle x2 + y2 = r2.The ratio of areas is therefore the same as the value of the determinant Δ = 2. Similarly, the inverse image of the circle u2 + υ2= R2 is given by the equation

Image

which is an ellipse bounding an area of π.R.R/2 = Image πR2. The ratio of areas is again equal to 2 (Fig. 14.7).

Image

FIGURE 14.6 Map F:(x, y) → (x, 2y); image of circle

Translations

There is another class of elementary transformations that is often considered together with the linear transformations. Namely, the transformation

Image

where h and k are constants. If we picture the x, y plane and u, υ plane as coinciding, then the effect of the transformation (14.18) is to move each point (x, y) a distance h horizontally and a distance k vertically. Equivalently,

Image

FIGURE 14.7Map F:(x, y)→ (x, 2y); inverse image of circle

each point undergoes a displacement represented by the fixed vector imageh, kimage Such a transformation is called a parallel translation, or simply a translation.

Combining a translation with a linear transformation, yields a transformation of the form

Image

A transformation of this form is called an affine transformation.

There is a certain ambiguity in the use of the word “linear,” which is sometimes used to mean “purely linear,” that is to say, homogeneous of first degree, and sometimes in the more general sense of a polynomial of degree one, which may include constant terms. Thus, in the case of functions, “linear” may mean ax+ by or ax + by + c. However, in the case of transformations it has become standard to reserve the expression “linear transformation” for the homogeneous case, that is, Eqs. (14.1), and to use the word “affine” for the more general transformations of the form (14.19).

An affine transformation can be most readily visualized as the result of two successive transformations. This is a special case of the following general situation, which we encounter frequently.

Definition 14.3 Let F1 be an arbitrary transformation of a domain D1 into a domain D2, and let F2 be a transformation of D2 into a domain D3. Then the transformation F of D1 into D3 obtained by applying successively the transformations F1 and F2 see Fig. 14.8) is called the composition of F1 and F2, and is denoted by

Image

Image

FIGURE 14.8Composed map

Note that the transformation written on the right is the one that is applied first. The reason for this convention is that if

Image

then

Image

so that the transformation applied last appears on the left.

In terms of coordinates, if

Image

then

Image

Thus, the general affine transformation (14.19) is obtained as the composition of

Image

where F1 is a general linear transformation and F2 is a translation. The equations for F = F2 image F1 are obtained by direct substitution into the equations for F2 of the expressions for X and Y given by F1.

In Sect. 15 we shall study in detail the composition of linear transformations, and in Sect. 17 we shall treat the composition of more general transformations.

Exercises

14.1Write down the matrices associated with the following linear transformations; compute the determinants, and decide whether the transformations are singular or nonsingular.

a. Image

b. Image

c. Image

d. Image

e. Image

f. Image

g. Image

h. Image

i. Image

j. Image

k. Image

l. Image

14.2Show that each of the following linear transformations has nonzero determinant Δ. Find the inverse transformation, compute its determinant Δ', and show that Δ' = 1/Δ.

a. Image

b. Image

c. Image

d. Image

14.3Show that each of the following linear transformations has zero determinant. Find the line in the u, υ plane that is the image of the whole plane under the transformation.

a. Image

b. Image

c. Image

d. Image

14.4 Consider the transformation

Image

a. What is the inverse image under F of a point on the line υ = 2u?

b. Show that F maps the line y = x in a one-to-one fashion onto the line υ = 2 u.

c. Show that each line perpendicular to y = x is mapped by F onto a single point.

d. Show that two different lines perpendicular to y = x are mapped by F onto two different points.

e. Deduce that the transformation F may be described geometrically in the following fashion. There is a family of parallel lines in the x, y plane such that F maps each line into a point on the line υ = 2u; as a variable line sweeps out the family of lines, the image under F describes the entire line υ = 2u.

14.5Let F be a singular linear transformation,

Image

where a and b are not both zero.

a. Show that there is a number λ such that F may be written in the form

Image

(Hint: see Ex. 1.9a.)

b. What is the inverse image of a point on the line υu?

c. Show that an arbitrary singular linear transformation F may be visualized geometrically in a fashion analogous to that described in Ex. 14.4e, unless F maps every point into the origin. Namely, there is a family of parallel lines in the x, y plane, such that F maps each line onto a point, F maps two different lines onto two different points, and as a variable line sweeps through the family, the image under F describes a line in the u, υ plane. (Hint: consider separately the case where a and b are both zero and the case where a and b are not both zero.)

14.6For each of the following linear transformations, introduce polar coordinates r, θ in the x, y plane and R, φ in the u, υ plane. Find φ explicitly as a function of θ, and verify that φ is an increasing function of θ when the determinant is positive, and a decreasing function of θ when the determinant is negative.

a. Image

b. Image

c. Image

d. Image

e. Image

14.7Let F be a nonsingular linear transformation.

a. Show that if F is orientation-preserving, then so is F−1.

b. Show that if F is orientation-reversing, then so is F−1.

14.8 For each of the transformations in Ex. 14.6, find the inverse image of u2 + υ2R2, find its area A, and show that the ratio of πR2/A is equal to the absolute value of the determinant.

14.9Let F be a linear transformation

Image

a. Show that if F(x0, y0) = (u0, υ0) and F(x1, y1) = (u1, υ1), then F maps the line segment joining (x0, y0) to (x1,y1)onto the line segment joining (u0, υ0) to (u1, υ1) (Hint: write the first line segment in parametric form, and substitute in the equations for F to get the equations of the image.)

b. Show that F maps every triangle onto a triangle, where the image triangle may be degenerate.

c. If T1is the triangle whose vertices are (0,0), (1,0), and (0,1), what are the vertices of the triangle T2, the image of T1 under F?

d. If A1, A2 are the areas of Tl, T2, respectively, show that A2/A1 = |Δ| = |adbc|. (Hint: see Ex. 1.29d.)

e. Show that the unit square, with vertices (0, 0), (1, 0), (1, 1), (0, 1), maps onto a parallelogram whose area is |adbc |.

14.10Let F be the transformation

Image

a. What is the shortest distance to the origin of points F(x, y), where (x, y) lies on the circle x2 + y2 =1?

(Hint : if d is the shortest distance, then d2 is the minimum of u2 + υ2 subject to the condition x2 + y2 = 1.)

b. What is the furthest distance to the origin of points F(x, y), where x2+ y2 =1?

c. Verify that the product of the answers to parts a and b is equal to the absolute value of the determinant of F, and explain why that is so.

*14.11Let F be an arbitrary linear transformation

Image

For any two points (x0, y0), (x1, y1), let (u0, υ0) = F(x0 y0) and (u1, υ1) = F(x1, y1).Consider the displacement vectors

Image

a. Show that F(X, Y) = (U, V).

b. Show that |image u, υimage |2 = AX2+ 2BXY + CY2, and determine the coefficients A, B, C.

c. Show that the quadratic form in part b is positive definite if and only if F is nonsingular, and positive semidefinite if and only if F is singular.

d. Using part c, give a direct proof that for a nonsingular linear transformation F, no two distinct points can map onto a single point (that is, F is injective), whereas for F singular, certain pairs of points map into a single point (that is, F is not injective).

e. Show that as (x0, y0) and (x1, y1) range over all pairs of distinct points in the x, y plane, the ratio |image u, υimage |/|image x, yimage | of the distance between the image points and the distance between the original points ranges between a minimum m and a maximum M, where the product mM is the absolute value of the determinant of F. In particular, if F is nonsingular, m > 0.

f. Show that for the transformation in Ex. 14.10, the quantities m and M of part e are m = 2, M = 4.

14.12Let F be an arbitrary linear transformation.

a. Show that F always maps the origin into the origin.

b. Show that if (x1, y1) is any point other than the origin, then specifying F(x1, y1) determines F(x, y) on the entire line through the origin and (x1, y1).

*c. Show that if (x1, y1) and (x2, y2) are any two points that do not lie on a straight line through the origin, then specifying F(x1,y1) and F(x2, y2) determines F(x, y) completely.

*d. Show that if (x1, y1) and (x2, y2) are any two points that do not lie on a straight line through the origin, and if (u1, u1), (u2, υ2) are any two points at all, then there exists one and only one linear transformation F such that F(x1, y1) = (u1, υ1) and F(x2, y2) = (u2, υ2).

e. Is the statement in part d still correct if (x1, y1) and (x2, y2) lie on a line through the origin? Explain.

14.13Prove the following statements.

a. Every parallel translation has an inverse, which is again a parallel translation.

b. The composition of two parallel translations is again a parallel translation.

c. If F1 and F2 are parallel translations, then F1 image F2 = F2 imageF1.

14.14Consider the linear transformations

Image

a. What is the image of (0, 1) under F2 image F1?

b. What is the image of (0, 1) under F1 image F2?

c. Show that F1 image F2F2 image F1.

d. Indicate with a sketch how the answers to parts a and b may be obtained geometrically, using the interpretations of F1 and F2 as transformations of a plane into itself.

*14.15Let F1 and F2 be linear transformations such that F2 image F1 leaves every point unchanged.

a. Show that F1 and F2 are both nonsingular.

b. Show that image

14.16A transformation F such that F image F leaves every point unchanged is called an involution.

a. Show that if a linear transformation F is an involution, then the determinant of Fis ± 1. (Hint: see Ex. 14.15.)

b. Which of the following transformations are involutions:

Image

*14.17Let F be a linear transformation

Image

where a, b, c, d are integers. Find a necessary and sufficient condition that the inverse transformation F−1 will have integer coefficients. Prove your assertion.

14.18Let F be an affine transformation

Image

a. Show that F maps every straight line segment onto a straight line segment or a point.

b. Under what condition does F have an inverse?

c. Write down the equations for F−1 when it exists.

14.19Let F1 map D1 into D2, and let F2 map D2 into D3.Show that if F1 and F2 have inverses, then so does F = F2 image F1, and image

Remark Given a matrix

Image

we may associate with it not only the transformation F of the plane defined by Eqs. (14.1), but also a corresponding transformation of vectors, which we may denote by F. Namely,

Image

(By Ex. 14.11a, F is precisely the transformation of displacement vectors induced by the transformation F of the plane.) In fact, it is more usual to consider linear transformations as acting upon vectors. The following five exercises explore linear transformations from this point of view. (Note that as transformations of number pairs into number pairs, F and F are identical. The only difference is in the geometric interpretation we give to these number pairs, and this difference only becomes essential when we consider allowable coordinate changes corresponding to the geometric interpretation.)

14.20Show that the transformation F defined above has the following properties.

a. Fw) = λF(w) for any vector w and any scalar λ.

b. F(w1 + w2) = F(w1) + F(w2) for any vectors w1, w2.

14.21Show that the properties in Ex. 14.20a, b are together equivalent to the single property

Image

for any vectors w1, w2 and scalars λ12.

14.22Let F be an arbitrary transformation that assigns to every vector w a vector F(w). Suppose that F satisfies Ex. 14.20a, b (or equivalently, the property in Ex. 14.21).

a. Show that F is completely determined by its values on image0,1 image and on image1,0image .

b. Show that F is a linear transformation.

14.23 a. Show that F(0) = 0.

b. Show that F(w) = 0 for some w 0 if and only if det image = 0.

* 14.24Definition: a nonzero vector w that is carried by F into a multiple of itself is called a characteristic vector of the transformation F and of the matrix image. (The terms proper vector and eigenvector are also used.) Thus, if w is a characteristic vector, F(w) = λw for some scalar A. This scalar λ is then called a characteristic value of F and of the matrix image.(Again, the expressions proper value and eigenvalue are used.)

a. Show that if λ is a characteristic value of the matrix image, then λ satisfies the equation

Image

or

Image

(Hint : write out the equation F(w) — λw = 0 in components, and use the fact that there exists a nonzero solution w.)

b. Show that a 2 × 2 matrix can have at most two characteristic values.

c. Show that the matrix image has no (real) characteristic values and hence no (real) characteristic vectors. Can you give a reason for this in terms of the geometric interpretation of the linear transformation

Image

(Hint: see Ex. 14.6e.)

d. Show that if det image< 0, then there always exist real nonzero characteristic values and characteristic vectors. Can you give a reason for this in terms of the geometric interpretation of the sign of the determinant?

e. Show that the maximum and minimum of the quadratic form AX2 + 2BXY + CY2,subject to the condition X2 + Y2 = 1, are the characteristic values of the matrix image furthermore if (X0, y0), (X1,Y1) are points where these maximum and minimum values are attained, then imageX0, Y0image and imageX1, Y1image are characteristic vectors of this matrix.

14.25Let C be the curve x = a cos t + b sin t, y = c cos t + d sin t, 0≤ t ≤ 2π.

a. Show that if adbc ≠ 0, then C is an ellipse.

b. Describe C for the case ad − bc = 0.