06. Linear Transformations Linear... · 2014-12-29 · 6.1. Matrices as Transformations All Linear Transformations from Rn to Rm Are Matrix Transformations The matrix A in this theorem

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6. Linear Transformations6. Linear Transformations


6.1. Matrices as Transformations6.1. Matrices as Transformations

A Review of Functions

http://en.wikipedia.org/wiki/Codomain

domain codomainrange

x y

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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”




If T is a transformation whose domain is Rn and whose range is in Rm, then we will write

You can think of a transformation T as mapping points into points or vectors into vectors.

(read, “T maps Rn into Rm”).

If T: RnRn, then we refer to the transformation T as an operator on Rn to emphasize that it maps Rn back into Rn.



Matrix Transformations

Matrix transformation

If A is an m×n matrix, and if x is a column vector in Rn, then the product Ax is a vector in Rm, so multiplying x by A creates a transformation that maps vectors in Rn into vectors in Rm. We call this transformation multiplication by A or the transformation A and denote it by TA to emphasize the matrix A.

and

or equivalently,

In the special case where A is square, say n×n, we have ,

and we call TA a matrix operator on Rn.

: n nAT R R



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,




If v1, v2, …, vk are vectors in Rn and c1, c2, …, ck are any scalars, then

Engineers and physicists sometimes call this the superposition principle.




Example 7Example 7From Theorem 3.1.5,

If A is an m×n matrix, u and v are column vectors in Rn, and c is a scalar, then A(cu)=c(Au) and A(u+v)=Au+Av.

Thus, the matrix transformation TA:RnRm is linear since



Some Properties of Linear Transformations



All Linear Transformations from Rn to Rm 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.







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)




REMARK

Theorem 6.1.4 shows that a linear transformation T:RnRm 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 11Example 11Show that the transformation T:R3R2 defined by the formula

is linear and find its standard matrix.



Rotations About The Origin

Let θ be a fixed angle, and consider the operator T that rotates each vector x in R2 about the origin through the angle θ.

T is linear.



Rotations About The Origin

We will denote the standard matrix for the rotation about the origin through an angle θ by Rθ.



Reflections About Lines Through The Origin

Let us consider the operator T:R2R2 that reflects each vector x about a line through the origin that makes an angle θ with the positive x-axis.









Orthogonal Projections onto Lines Through The Origin

Consider the operator T:R2R2 that projects each vector x in R2 onto a line through the origin by dropping a perpendicular to that line.




The standard matrix for an orthogonal projection onto a general line through the origin can be obtained using Theorem 6.1.4.

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 Pθ.

Solving for Pθx yeilds

so part (b) of Theorem 3.4.4 implies that




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.

Example 14Example 14

3 11 142 2 4

/1231 1 1

4 2 2 4

3 31 1 1.181 0.321 3 3

P

x






Transformations of The Unit Square

The unit square in R2 is the square that has e1 and e2 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 R2 by graphing the images of these vertices.


6.2. Geometry of Linear Operators6.2. Geometry of Linear Operators

Norm-Preserving Linear Operators

Length preserving, angle preserving

A linear operator T:RnRn with the length-preserving property ||T(x)||=||x|| is called an orthogonal operator or a linear isometry (from the Greek isometros, meaning “equal measure”).




(a)(b) Suppose that T is length preserving, and let x and y be any two vectors in Rn.

(4)




(b)(a) Conversely, suppose that T is dot product preserving, and let x be any vector in Rn.

Since

It follows that



Orthogonal Operators Preserve Angles And Orthogonality

Recall from the remark following Theorem 1.2.12 that the angle between two nonzero vectors x and y in Rn is given by the formula

Thus, if T:RnRn 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.



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:RnRn. Since T(x)=Ax for all x in Rn, and since ||T(x)||=||x||, it follows that

for all x in Rn.

2 2

T T

T T T

A

A A

A A

A A

x x

x x x x

x x x x

x x x x

TA A I 1 TA A



Orthogonal Matrices

The matrix

Example 1Example 1

is orthogonal since



Orthogonal Matrices

and hence

Example 1Example 1



Orthogonal Matrices



Orthogonal Matrices



Orthogonal Matrices

In the case of a square matrix, Theorem 6.2.4 and 6.2.3 together yield the following theorem about orthogonal matrices.



Orthogonal Matrices

Example 2Example 2



All Orthogonal Linear Operators on R2 Are Rotations or Reflections



All Orthogonal Linear Operators on R2 Are Rotations or Reflections

If A is an orthogonal 2×2 matrix, then we know from Theorem 6.2.7 that the correspondinglinear 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.



Contractions and Dilations of R2

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.



Vertical and Horizontal Compressions And Expansions of R2

Expansion (or compression) in the x-direction with factor kT(x,y)=(kx,y)



Vertical and Horizontal Compressions And Expansions of R2

Expansion (or compression) in the y-direction with factor kT(x,y)=(x,ky)



Shears

Shear in the x-direction with factor kT(x,y)=(x+ky,y)

T(x,y)=(x,y+kx)Shear in the y-direction with factor k



Shears

Example 5Example 5



Shears

Example 6Example 6



Linear Operators on R3

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 xyz-coordinate system.



Linear Operators on R3

All 3×3 orthogonal matrices correspond to linear operators on R3 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.



Reflections about Coordinate Planes



Rotations in R3

A rotation of R3 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:R3R3 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)



Rotations in R3



General Rotations



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



General Rotations

(a) Show that the matrixExample 7Example 7

represents a rotation about a line through the origin of R3.

(b) Find the axis and angle of rotation.



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

If A is a rotation matrix, then for any nonzero vector x in R3 that is not perpendicular to the axis of rotation, the vector

is nonzero and is along the axis of rotation when x has its initial point at the origin.

(16)

(17)



General Rotations

Use Formulas (16) and (17) to solve the problem in part (b) of Example 7.Example 8Example 8

(16)

(17)


6.3. Kernel and Range6.3. Kernel and Range

Kernel of A Linear Transformation

In each part, find the kernel of the stated linear operator on R3.

(a) The zero operator T0(x)=0x=0: kel(T0)=R3

(b) The identity operator TI(x)=Ix=x: ker(TI)={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}

Example 1Example 1

It is important to note that the kernel of a linear transformation always contains the vector 0 by Theorem 6.1.3.



Kernel of A Linear Transformation

The kernel of T is a nonempty set since it contains the zero vector in Rn.

To show that it is a subspace of Rn 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.



Kernel of A Matrix Transformation




Find the null space of the matrix

Example 3Example 3

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{v1, v2, v3}




Let S be any subspace of Rn, and let W=T(S) be its image under T.

Suppose that u and v are the images of the vector u0 and v0 in S, respectively; that is,

Since S is a subspace of Rn, it is closed under scalar multiplication and addition, so cu0 and u0+v0 are also vectors in S.

and

and

which shows that cu and u+v are images of vectors in S.

Thus, W is closed under scalar multiplication and addition.



Range of A Linear Transformation

The range of a linear transformation T:RnRm can be viewed as the image of Rn under T, so it follows as a special case of Theorem 6.3.5 that the range of T is a subspace of Rm.



Range of A Matrix Transformation

If TA:RnRm is the linear transformation corresponding to the matrix A, then the range of TAand the column space of A are the same object from different points of view – the first emphasizes the transformation and the second the matrix.



Range of A Matrix Transformation

It is important in many kinds of problems to be able to determine whether a given vector b in Rm is in the range of a linear transformation T:RnRm. If A is the standard matrix for T, then this problem reduces to determining whether b is in the column space of A.

Example 6Example 6Suppose 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.



Existence and Uniqueness Issues




The rotation is onto and one-to-one.Example 7Example 7

Example 8Example 8

The orthogonal projection is neither onto nor one-to-one.

Example 9Example 9

T(x,y)=(x,y,0) is one-to-one, but is not onto.


T(x,y,z)=(x,y) is onto, but is not one-to-one.




(a)(b)

Since T is linear, T(0)=0 by Theorem 6.1.3. 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 x1≠x2, then x1-x2≠0, which means that x1-x2 is not in ker(T). This being the case,

Thus, T(x1)≠T(x2).







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 Rn.

Combining this with Theorem 6.3.12 and 6.3.13 completes the proof.



A Unifying Theorem



The fact that a rotation about the origin R2 is one-to-one and onto can be established algebraically by showing that the determinant of its matrix is not zero.

A Unifying Theorem


The fact that the orthogonal projection of R3 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.


6.4. Composition and Invertibility6.4. Composition and Invertibility

Compositions of Linear Transformations

First applying T1 and then applying T2 to the output of T1 produces a transformation from Rn to Rm. This transformation, called the composition of T2 with T1, is denoted by T2◦T1 (read, “T2circle T1”)




Let u and v be any vectors in Rn, and let c be a scalar.







Example 1Example 1




Example 2Example 2

We see that this matrix represents a rotation about the origin through an angle of 2θ2-2θ1.




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.




REMARK

The composition of linear operators is the same in either order if and only if their standard matrices commute.



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 (T3◦T2◦T1):RnRm by




Let A1, A2, …, Ak 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 R3. Since A is the standard matrix for the composition Tk◦T2◦T1, the result is proved.




In aeronautics and astronautics, the orientation of an aircraft or space shuttle relative to an xyz-coordinate 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.




Suppose that a vector in R3 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.

Example 5Example 5




To find the axis of rotation v we will apply Formula (17) of Section 6.2, taking the arbitrary vector x to be e1.

Example 5Example 5

(16)

(17)

Also, it follows from Formula (16) of Section 6.2 that the angle of rotation satisfies

from which it follows that θ ≈ 64.74º



Factoring Linear Operators into Compositions

A diagonal matrixExample 6Example 6

can be factored as




If

has nonnegative entries, then multiplication by D maps the standard unit vector ei into the vector λiei, 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.




The 2×2 elementary matrices have five possible forms:Example 7Example 7

If k<0, then we can express k in the form k=-k1, where k1>0, and we can factor the type 4 and 5 matrices as




Recall from Theorem 3.3.3 that an invertible matrix A can be expressed as a product of elementary matrices.

Example 8Example 8

Describe the geometric effect of multiplication by

in terms of shears, compression, expansions, and reflections.



Inverse of A Linear Transformation

T-1(w)=x if and only if T(x)=wThis function is called inverse of T.

T: RnRm

(13)



Invertible Linear Operators

Let A and B be the standard matrices for T and T-1, respectively, and let x be any vector in Rn. We know from (13) that

which we can write in matrix form as

Since this holds for all x in Rn, it follows from Theorem 3.4.4 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.



Invertible Linear Operators

Example 9Example 9




Geometric Properties of Invertible Linear Operators on R2



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 6.4.7.


6.5. Computer Graphics6.5. Computer Graphics

Wireframes, Matrix Representations of Wireframes

Example 1Example 1

vertices connectivity matrix

wire

wireframe



Transforming Wireframes

The connectivity matrix does not change.

Example 3Example 3

Shearing the roman version in the position x-direction to an angle that is 15º off the vertical.



Translation Using Homogeneous Coordinates

If x=(x1,x2,…,xn) is a vector in Rn, then the modified vector (x1,x2,…,xn,1) in Rn+1 is said to represent x in 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.

Example 4Example 4




Example 5Example 5




Example 6Example 6




Example 7Example 7



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.




orthogonal projectionperspective projection

vanishing point




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.

Example 8Example 8




Example 8Example 8




Example 8Example 8

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06. Linear Transformations Linear... · 2014-12-29 · 6.1. Matrices as Transformations All Linear Transformations from Rn to Rm Are Matrix Transformations The matrix A in this theorem