6. Linear Transformations.

Transcription

1 6. Linear Transformations

2 6.1. Matrices as Transformations A Review of Functions domain codomain range x y preimage image

3 6.1. Matrices as Transformations A Review of Functions A function whose input and outputs are vectors is called a transformation, and it is standard to denote transformations by capital letters such as F, T, or L. w=t(x) T maps x into w

4 6.1. Matrices as Transformations A Review of Functions If T is a transformation whose domain is R n and whose range is in R m, then we will write (read, T maps R n into R m ). You can think of a transformation T as mapping points into points or vectors into vectors. If T: R n R n, then we refer to the transformation T as an operator on R n to emphasize that it maps R n back into R n.

5 6.1. Matrices as Transformations Matrix Transformations Matrix transformation If A is an m n matrix, and if x is a column vector in R n, then the product Ax is a vector in R m, so multiplying x by A creates a transformation that maps vectors in R n into vectors in R m. We call this transformation multiplication by A or the transformation A and denote it by T A to emphasize the matrix A. and or equivalently, n n In the special case where A is square, say n n, we have T : R R, and we call T A a matrix operator on R n. A

6 6.1. Matrices as Transformations Linear Transformations The operational interpretation of linearity 1. Homogeneity: Changing the input by a multiplicative factor changes the output by the same factor; that is, 2. Additivity: Adding two inputs adds the corresponding outputs; that is,

7 6.1. Matrices as Transformations Linear Transformations If v 1, v 2,, v k are vectors in R n and c 1, c 2,, c k are any scalars, then Engineers and physicists sometimes call this the superposition principle.

8 6.1. Matrices as Transformations Linear Transformations Example 7 From Theorem 3.1.5, If A is an m n matrix, u and v are column vectors in R n, and c is a scalar, then A(cu)=c(Au) and A(u+v)=Au+Av. Thus, the matrix transformation T A :R n R m is linear since

9 6.1. Matrices as Transformations Some Properties of Linear Transformations

10 6.1. Matrices as Transformations All Linear Transformations from R n to R m Are Matrix Transformations The matrix A in this theorem is called the standard matrix for T, and we say that T is the transformation corresponding to A, or that T is the transformation represented by A, or sometimes simply that T is the transformation A.

11 6.1. Matrices as Transformations All Linear Transformations from R n to R m Are Matrix Transformations

12 6.1. Matrices as Transformations All Linear Transformations from R n to R m Are Matrix Transformations When it is desirable to emphasize the relationship between T and its standard matrix, we will denote A by [T]; that is, we will write With this notation, the relation ship in (13) becomes (14) (13)

13 6.1. Matrices as Transformations All Linear Transformations from R n to R m Are Matrix Transformations REMARK Theorem shows that a linear transformation T:R n R m is completely determined by its values at the standard unit vectors in the sense that once the images of the standard unit vectors are known, the standard matrix [T] can be constructed and then used to compute images of all other vectors using (14) Example 11 Show that the transformation T:R 3 R 2 defined by the formula is linear and find its standard matrix.

14 6.1. Matrices as Transformations Rotations About The Origin Let θ be a fixed angle, and consider the operator T that rotates each vector x in R 2 about the origin through the angle θ. T is linear.

15 6.1. Matrices as Transformations Rotations About The Origin We will denote the standard matrix for the rotation about the origin through an angle θ by R θ.

16 6.1. Matrices as Transformations Reflections About Lines Through The Origin Let us consider the operator T:R 2 R 2 that reflects each vector x about a line through the origin that makes an angle θ with the positive x-axis.

17 6.1. Matrices as Transformations Reflections About Lines Through The Origin

18 6.1. Matrices as Transformations Reflections About Lines Through The Origin

19 6.1. Matrices as Transformations Orthogonal Projections onto Lines Through The Origin Consider the operator T:R 2 R 2 that projects each vector x in R 2 onto a line through the origin by dropping a perpendicular to that line.

20 6.1. Matrices as Transformations Orthogonal Projections onto Lines Through The Origin The standard matrix for an orthogonal projection onto a general line through the origin can be obtained using Theorem Consider a line through the origin that makes an angle θ with the positive x-axis, and denote the standard matrix for the orthogonal projection by P θ. Solving for P θ x yeilds so part (b) of Theorem implies that

21 6.1. Matrices as Transformations Orthogonal Projections onto Lines Through The Origin Example 14 Find the orthogonal projectin of the vector x=(1,1) on the line through the origin that makes an angle of π/12(=15º) with the x-axis P /12x

22 6.1. Matrices as Transformations Orthogonal Projections onto Lines Through The Origin

23 6.1. Matrices as Transformations Transformations of The Unit Square The unit square in R 2 is the square that has e 1 and e 2 as adjacent side; its vertices are (0,0), (1,0), (1,1), and (0,1). It is often possible to gain some insight into the geometric behavior of a linear operator on R 2 by graphing the images of these vertices.

24 6.2. Geometry of Linear Operators Norm-Preserving Linear Operators Length preserving, angle preserving A linear operator T:R n R n with the length-preserving property T(x) = x is called an orthogonal operator or a linear isometry (from the Greek isometros, meaning equal measure ).

25 6.2. Geometry of Linear Operators Norm-Preserving Linear Operators (a) (b) Suppose that T is length preserving, and let x and y be any two vectors in R n. (4)

26 6.2. Geometry of Linear Operators Norm-Preserving Linear Operators (b) (a) Conversely, suppose that T is dot product preserving, and let x be any vector in Rn. Since It follows that

27 6.2. Geometry of Linear Operators Orthogonal Operators Preserve Angles And Orthogonality Recall from the remark following Theorem that the angle between two nonzero vectors x and y in R n is given by the formula Thus, if T:R n R n is an orthogonal operator, the fact that T is length preserving and dot product preserving implies that which implies that an orthogonal operator preserves angles.

28 6.2. Geometry of Linear Operators Orthogonal Matrices Our next goal is to explore the relationship between the orthogonality of an operator and properties of its standard matrix. Suppose that A is the standard matrix for an orthogonal linear operator T:R n R n. Since T(x)=Ax for all x in R n, and since T(x) = x, it follows that for all x in R n. Ax 2 2 x A A x x x x T A A T x x x x T A A I A A 1 T AA T T T x x x x

29 6.2. Geometry of Linear Operators Orthogonal Matrices Example 1 The matrix is orthogonal since

30 6.2. Geometry of Linear Operators Orthogonal Matrices Example 1 and hence

31 6.2. Geometry of Linear Operators Orthogonal Matrices

32 6.2. Geometry of Linear Operators Orthogonal Matrices

33 6.2. Geometry of Linear Operators Orthogonal Matrices In the case of a square matrix, Theorem and together yield the following theorem about orthogonal matrices.

34 6.2. Geometry of Linear Operators Orthogonal Matrices Example 2

35 6.2. Geometry of Linear Operators All Orthogonal Linear Operators on R 2 Are Rotations or Reflections

36 6.2. Geometry of Linear Operators All Orthogonal Linear Operators on R 2 Are Rotations or Reflections If A is an orthogonal 2 2 matrix, then we know from Theorem that the corresponding linear operator is either a rotation about the origin or a reflection about a line through the origin. The determinant of A can be used to distinguish between the two cases, since it follows from (1) and (2) that Thus, a 2 2 orthogonal matrix represents a rotation if det(a)=1 and a reflection if det(a)=-1.

37 6.2. Geometry of Linear Operators Contractions and Dilations of R 2 If k is a nonnegative scalar, then the linear operator T(x,y)=(kx,ky) is called the scaling operator with factor k. In particular, this operator is called a contractor if 0 k<1 and a dilation if k>1.

38 6.2. Geometry of Linear Operators Vertical and Horizontal Compressions And Expansions of R 2 T(x,y)=(kx,y) Expansion (or compression) in the x-direction with factor k

39 6.2. Geometry of Linear Operators Vertical and Horizontal Compressions And Expansions of R 2 T(x,y)=(x,ky) Expansion (or compression) in the y-direction with factor k

40 6.2. Geometry of Linear Operators Shears T(x,y)=(x+ky,y) Shear in the x-direction with factor k T(x,y)=(x,y+kx) Shear in the y-direction with factor k

41 6.2. Geometry of Linear Operators Shears Example 5

42 6.2. Geometry of Linear Operators Shears Example 6

43 6.2. Geometry of Linear Operators Linear Operators on R 3 The most important linear operators that are not length preserving are orthogonal projections onto subspaces, and the simplest of these are the orthogonal projections onto the coordinate planes of xyzcoordinate system.

44 6.2. Geometry of Linear Operators Linear Operators on R 3 All 3 3 orthogonal matrices correspond to linear operators on R 3 of the following types: Type 1: Rotations about lines through the origin Type 2: Reflections about planes through the origin Type 3: A rotation about a line through the origin followed by a reflection about the plane through the origin that is perpendicular to the line If A is a 3 3 orthogonal matrix, then A represents a rotations (i.e., is of type 1) if det(a)=1 and represents a type 2 or type 3 operator if det(a)=-1. Accordingly, we will frequently refer to 2 2 or 3 3 orthogonal matrices with determinant 1 as rotation matrices.

45 6.2. Geometry of Linear Operators Reflections about Coordinate Planes

46 6.2. Geometry of Linear Operators Rotations in R 3 A rotation of R 3 is an orthogonal operator with a line of fixed points, called the axis of rotation. In this section we will only be concerned with rotations about lines through the origin, and we will assume for simplicity that an angle of rotation is at most 180. If T:R 3 R 3 is a rotation through an angle θ about a line through the origin, and if W is the plane through the origin that is perpendicular to the axis of rotation, then T rotates each nonzero vector w in W about the origin through the angle θ into a vector T(w) in W. The orientation of the axis of rotation u=w T(w)

47 6.2. Geometry of Linear Operators Rotations in R 3

48 6.2. Geometry of Linear Operators General Rotations

49 6.2. Geometry of Linear Operators General Rotations Given the standard matrix for a rotation, find the axis and angle of rotation. Since the axis of rotation consists of the fixed points of A, we can determine this axis by solving the linear system Once we know the axis of rotation, we can find a nonzero vector w in the plane W through the origin that is perpendicular to this axis and orient the axis using the vector Looking toward the origin from the terminal point of u, the angle of rotation will be counterclockwise in W and hence can be computed from the formula

50 6.2. Geometry of Linear Operators General Rotations Example 7 (a) Show that the matrix represents a rotation about a line through the origin of R 3. (b) Find the axis and angle of rotation.

51 6.2. Geometry of Linear Operators General Rotations A formula for the cosine of the rotation angle in terms of the entries of A can be obtained from (13) by observing that from which it follows that (16) If A is a rotation matrix, then for any nonzero vector x in R 3 that is not perpendicular to the axis of rotation, the vector (17) is nonzero and is along the axis of rotation when x has its initial point at the origin.

52 6.2. Geometry of Linear Operators General Rotations Example 8 Use Formulas (16) and (17) to solve the problem in part (b) of Example 7. (16) (17)

53 6.3. Kernel and Range Kernel of A Linear Transformation Example 1 In each part, find the kernel of the stated linear operator on R 3. (a) The zero operator T 0 (x)=0x=0: kel(t 0 )=R 3 (b) The identity operator T I (x)=ix=x: ker(t I )={0} (c) The orthogonal projection T on the xy-plane: ker(t)=z-axis (d) A rotation T about a line through the origin through an angle θ: ker(t)={0} It is important to note that the kernel of a linear transformation always contains the vector 0 by Theorem

54 6.3. Kernel and Range Kernel of A Linear Transformation The kernel of T is a nonempty set since it contains the zero vector in R n. To show that it is a subspace of R n we must show that it is closed under scalar multiplication and addition. Let u and v be any vectors in ker(t), and let c be any scalar.

55 6.3. Kernel and Range Kernel of A Matrix Transformation

56 6.3. Kernel and Range Kernel of A Matrix Transformation Example 3 Find the null space of the matrix In Example 7 of Section 2.2, where we showed that the solution space consist of all linear combinations of the vectors Thus, null (A)=span{v 1, v 2, v 3 }

57 6.3. Kernel and Range Kernel of A Matrix Transformation Let S be any subspace of R n, and let W=T(S) be its image under T. Suppose that u and v are the images of the vector u 0 and v 0 in S, respectively; that is, and Since S is a subspace of R n, it is closed under scalar multiplication and addition, so cu 0 and u 0 +v 0 are also vectors in S. and which shows that cu and u+v are images of vectors in S. Thus, W is closed under scalar multiplication and addition.

58 6.3. Kernel and Range Range of A Linear Transformation The range of a linear transformation T:R n R m can be viewed as the image of R n under T, so it follows as a special case of Theorem that the range of T is a subspace of R m.

59 6.3. Kernel and Range Range of A Matrix Transformation If T A :R n R m is the linear transformation corresponding to the matrix A, then the range of T A and the column space of A are the same object from different points of view the first emphasizes the transformation and the second the matrix.

60 6.3. Kernel and Range Range of A Matrix Transformation It is important in many kinds of problems to be able to determine whether a given vector b in R m is in the range of a linear transformation T:R n R m. If A is the standard matrix for T, then this problem reduces to determining whether b is in the column space of A. Example 6 Suppose that Determine whether b is in the column space of A, and, if so, express it as a linear combination of the column vectors of A.

61 6.3. Kernel and Range Existence and Uniqueness Issues

62 6.3. Kernel and Range Existence and Uniqueness Issues Example 7 The rotation is onto and one-to-one. Example 8 The orthogonal projection is neither onto nor one-to-one. Example 9 T(x,y)=(x,y,0) is one-to-one, but is not onto. Example 10 T(x,y,z)=(x,y) is onto, but is not one-to-one.

63 6.3. Kernel and Range Existence and Uniqueness Issues (a) (b) Since T is linear, T(0)=0 by Theorem The fact that T is one-to-one implies that x=0 is the only vector for which T(x)=0, so ker(t)={0}. (b) (a) If x 1 x 2, then x 1 -x 2 0, which means that x 1 -x 2 is not in ker(t). This being the case, Thus, T(x 1 ) T(x 2 ).

64 6.3. Kernel and Range Existence and Uniqueness Issues

65 6.3. Kernel and Range Existence and Uniqueness Issues Let A be the standard matrix for T. By parts (d) and (e) of Theorem 4.4.7, the system Ax=0 has only trivial solution if and only if the system Ax=b is consistent for every vector b in R n. Combining this with Theorem and completes the proof.

66 6.3. Kernel and Range A Unifying Theorem

67 6.3. Kernel and Range A Unifying Theorem Example 13 The fact that a rotation about the origin R 2 is one-to-one and onto can be established algebraically by showing that the determinant of its matrix is not zero. The fact that the orthogonal projection of R 3 on the xy-plane is neither one-to-one nor onto can be established by showing that the determinant of its standard matrix A is zero.

68 6.4. Composition and Invertibility Compositions of Linear Transformations First applying T 1 and then applying T 2 to the output of T 1 produces a transformation from R n to R m. This transformation, called the composition of T 2 with T 1, is denoted by T 2 T 1 (read, T 2 circle T 1 )

69 6.4. Composition and Invertibility Compositions of Linear Transformations Let u and v be any vectors in R n, and let c be a scalar.

70 6.4. Composition and Invertibility Compositions of Linear Transformations

71 6.4. Composition and Invertibility Compositions of Linear Transformations Example 1

72 6.4. Composition and Invertibility Compositions of Linear Transformations Example 2 We see that this matrix represents a rotation about the origin through an angle of 2θ 2-2θ 1.

73 6.4. Composition and Invertibility Compositions of Linear Transformations REMARK From Theorem 6.2.7, since a rotation is represented by an orthogonal matrix with determinant +1 and a reflection by an orthogonal matrix with determinant -1, the product of two rotation matrices or two reflections is an orthogonal matrix with determinant +1 and hence represents a rotation.

74 6.4. Composition and Invertibility Compositions of Linear Transformations REMARK The composition of linear operators is the same in either order if and only if their standard matrices commute.

75 6.4. Composition and Invertibility Composition of Three or More Linear Transformations Compositions can be defined for three or more matrix transformations when the domains and codomains match up appropriately. Specially, if then we define the composition (T 3 T 2 T 1 ):R n R m by

76 6.4. Composition and Invertibility Composition of Three or More Linear Transformations Let A 1, A 2,, A k be the standard matrices for the rotations. Each matrix is orthogonal and has determinant 1, so the same is true for the product Thus, A represents a rotation about some axis through the origin of R 3. Since A is the standard matrix for the composition T k T 2 T 1, the result is proved.

77 6.4. Composition and Invertibility Composition of Three or More Linear Transformations In aeronautics and astronautics, the orientation of an aircraft or space shuttle relative to an xyzcoordinate system is often described in terms of angles called yaw, pitch, and roll. As a result of Theorem 6.4.3, a combination of yaw, pitch, and roll can be achieved by a single rotation about some axis through the origin. This is, in fact, how a space shuttle makes attitude adjustments it doesn t perform each rotation separately; it calculates one axis, and rotates about that axis to get the correct orientation.

78 6.4. Composition and Invertibility Composition of Three or More Linear Transformations Example 5 Suppose that a vector in R 3 is first rotated 45º about the positive x-axis, then the resulting vector is rotated 45º about the positive y-axis, and then that vector is rotated 45º about the positive z-axis. Find an appropriate axis and angle of rotation that achieves the same result in one rotation.

79 6.4. Composition and Invertibility Composition of Three or More Linear Transformations Example 5 To find the axis of rotation v we will apply Formula (17) of Section 6.2, taking the arbitrary vector x to be e 1. (17) Also, it follows from Formula (16) of Section 6.2 that the angle of rotation satisfies (16) from which it follows that θ 64.74º

80 6.4. Composition and Invertibility Factoring Linear Operators into Compositions Example 6 A diagonal matrix can be factored as

81 6.4. Composition and Invertibility Factoring Linear Operators into Compositions If has nonnegative entries, then multiplication by D maps the standard unit vector e i into the vector λ i e i, so you can think of this operator as causing compressions or expansions in the directions of the standard unit vectors. Because of these geometric properties, diagonal matrices with nonnegative entries are called scaling matrices.

82 6.4. Composition and Invertibility Factoring Linear Operators into Compositions Example 7 The 2 2 elementary matrices have five possible forms: If k<0, then we can express k in the form k=-k 1, where k 1 >0, and we can factor the type 4 and 5 matrices as

83 6.4. Composition and Invertibility Factoring Linear Operators into Compositions Recall from Theorem that an invertible matrix A can be expressed as a product of elementary matrices. Example 8 Describe the geometric effect of multiplication by in terms of shears, compression, expansions, and reflections.

84 6.4. Composition and Invertibility Inverse of A Linear Transformation T -1 (w)=x if and only if T(x)=w This function is called inverse of T. T: R n R m (13)

85 6.4. Composition and Invertibility Invertible Linear Operators Let A and B be the standard matrices for T and T -1, respectively, and let x be any vector in R n. We know from (13) that which we can write in matrix form as Since this holds for all x in R n, it follows from Theorem that BA=I. Thus, A is invertible and its inverse is B, which is what we wanted to prove. A one-to-one linear operator is also called an invertible linear operator.

86 6.4. Composition and Invertibility Invertible Linear Operators Example 9 Example 11

87 6.4. Composition and Invertibility Geometric Properties of Invertible Linear Operators on R 2

88 6.4. Composition and Invertibility Image of The Unit Square under An Invertible Linear Operator Since a linear operator maps 0 into 0, the vertex at the origin remains fixed under the transformation. The images of the other three vertices must be distinct, for otherwise they would lie on a line, and this is impossible by part (d) of Theorem

89 6.5. Computer Graphics Wireframes, Matrix Representations of Wireframes Example 1 wire wireframe vertices connectivity matrix

90 6.5. Computer Graphics Transforming Wireframes Example 3 Shearing the roman version in the position x-direction to an angle that is 15º off the vertical. The connectivity matrix does not change.

91 6.5. Computer Graphics Translation Using Homogeneous Coordinates Although translation is an important operation in computer graphics, it presents a problem because it is not a linear operator and hence not a matrix operator. If x=(x 1,x 2,,x n ) is a vector in R n, then the modified vector (x 1,x 2,,x n,1) in R n+1 is said to represent x in homogeneous coordinates. Example 4

92 6.5. Computer Graphics Translation Using Homogeneous Coordinates Example 5

93 6.5. Computer Graphics Translation Using Homogeneous Coordinates Example 6

94 6.5. Computer Graphics Translation Using Homogeneous Coordinates Example 7

95 6.5. Computer Graphics Three-Dimensional Graphics If, as in Figure 6.5.7, we imagine a viewer s eye to be positioned at a point Q(0,0,d) on the z- axis, then a vertex P(x,y,z) of the wireframe can be represented on the computer screen by the point (x*,y*,0) at which the ray from Q through P intersects the screen. These are called the screen coordinates of P, and this procedure for obtaining screen coordinates is called the perspective projection with viewpoint Q.

96 6.5. Computer Graphics Three-Dimensional Graphics vanishing point perspective projection orthogonal projection

97 6.5. Computer Graphics Three-Dimensional Graphics Example 8 The vertex matrix for the orthogonal projection of the rotated wireframe on the xy-plane can be obtained by setting the z-coordinates equal to zero.

98 6.5. Computer Graphics Three-Dimensional Graphics Example 8

99 6.5. Computer Graphics Three-Dimensional Graphics Example 8