# 06 Linear Transformations ```6. Linear Transformations
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6.1. Matrices as Transformations
A Review of Functions
domain
codomain
range
xy
preimage image
http://en.wikipedia.org/wiki/Codomain
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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”
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6.1. Matrices as Transformations
A Review of Functions
If T is a transformation whose domain is Rn and whose range is in Rm, then we will write
(read, “T maps Rn into Rm”).
You can think of a transformation T as mapping points into points or vectors into vectors.
If T: RnRn, then we refer to the transformation T as an operator on Rn to emphasize that it
maps Rn back into Rn.
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6.1. Matrices as Transformations
Matrix Transformations
Matrix transformation
If A is an m&times;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&times;n, we have TA : R n  R n ,
and we call TA a matrix operator on Rn.
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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,
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6.1. Matrices as Transformations
Linear Transformations
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.
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6.1. Matrices as Transformations
Linear Transformations
Example 7
From Theorem 3.1.5,
If A is an m&times;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:RnRm is linear since
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6.1. Matrices as Transformations
Some Properties of Linear Transformations
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6.1. Matrices as 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.
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6.1. Matrices as Transformations
All Linear Transformations from Rn to Rm Are Matrix Transformations
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6.1. Matrices as Transformations
All Linear Transformations from Rn to Rm 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)
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6.1. Matrices as Transformations
All Linear Transformations from Rn to Rm Are Matrix Transformations
REMARK
Theorem 6.1.4 shows that a linear transformation T:RnRm 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:R3R2 defined by the formula
is linear and find its standard matrix.
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6.1. Matrices as Transformations
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.
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6.1. Matrices as Transformations
We will denote the standard matrix for the rotation about the origin through an angle θ by Rθ.
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6.1. Matrices as Transformations
Reflections About Lines Through The Origin
Let us consider the operator T:R2R2 that reflects each vector x about a line through the
origin that makes an angle θ with the positive x-axis.
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6.1. Matrices as Transformations
Reflections About Lines Through The Origin
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6.1. Matrices as Transformations
Reflections About Lines Through The Origin
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6.1. Matrices as Transformations
Orthogonal Projections onto Lines Through The Origin
Consider the operator T:R2R2 that projects each vector x in R2 onto a line through the origin
by dropping a perpendicular to that line.
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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 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
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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(=15&ordm;) with the x-axis.

 12 1 
P /1 2 x  

1
4

3
2

1
4
1
2
1  
3
2






1
1   4 3 
1    1
  4 3



3   1 .1 8 

 0 .3 2 


3  

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6.1. Matrices as Transformations
Orthogonal Projections onto Lines Through The Origin
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6.1. Matrices as Transformations
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.
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6.2. Geometry of Linear Operators
Norm-Preserving Linear Operators
Length preserving, angle preserving
A linear operator T:RnRn with the length-preserving property ||T(x)||=||x|| is called an
orthogonal operator or a linear isometry (from the Greek isometros, meaning “equal
measure”).
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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 Rn.
(4)
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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
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6.2. Geometry of Linear Operators
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:RnRn 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.
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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:RnRn. Since
T(x)=Ax for all x in Rn, and since ||T(x)||=||x||, it follows that
for all x in Rn.
Ax  x
2
2
 Ax    Ax   x  x
T
T
A
A
x
x

x
    x
AT A  I
A1  AT
xT AT Ax  xT x
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6.2. Geometry of Linear Operators
Orthogonal Matrices
Example 1
The matrix
is orthogonal since
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6.2. Geometry of Linear Operators
Orthogonal Matrices
Example 1
and hence
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6.2. Geometry of Linear Operators
Orthogonal Matrices
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6.2. Geometry of Linear Operators
Orthogonal Matrices
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6.2. Geometry of Linear Operators
Orthogonal Matrices
In the case of a square matrix, Theorem 6.2.4 and 6.2.3 together yield the following theorem
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6.2. Geometry of Linear Operators
Orthogonal Matrices
Example 2
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6.2. Geometry of Linear Operators
All Orthogonal Linear Operators on R2 Are Rotations or Reflections
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6.2. Geometry of Linear Operators
All Orthogonal Linear Operators on R2 Are Rotations or Reflections
If A is an orthogonal 2&times;2 matrix, then we know from Theorem 6.2.7 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&times;2 orthogonal matrix represents a rotation if det(A)=1 and a reflection if det(A)=-1.
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6.2. Geometry of Linear Operators
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&lt;1 and a
dilation if k&gt;1.
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6.2. Geometry of Linear Operators
Vertical and Horizontal Compressions And Expansions of R2
T(x,y)=(kx,y)
Expansion (or compression) in the x-direction with factor k
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6.2. Geometry of Linear Operators
Vertical and Horizontal Compressions And Expansions of R2
T(x,y)=(x,ky)
Expansion (or compression) in the y-direction with factor k
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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
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6.2. Geometry of Linear Operators
Shears
Example 5
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6.2. Geometry of Linear Operators
Shears
Example 6
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6.2. Geometry of Linear Operators
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 xyzcoordinate system.
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6.2. Geometry of Linear Operators
Linear Operators on R3
All 3&times;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&times;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&times;2 or 3&times;3 orthogonal matrices with determinant 1
as rotation matrices.
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6.2. Geometry of Linear Operators
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6.2. Geometry of Linear Operators
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:R3R3 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&times;T(w)
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6.2. Geometry of Linear Operators
Rotations in R3
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6.2. Geometry of Linear Operators
General Rotations
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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
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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 R3.
(b) Find the axis and angle of rotation.
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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 R3 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.
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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)
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6.3. Kernel and Range
Kernel of A Linear Transformation
Example 1
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}
It is important to note that the kernel of a linear transformation always contains the
vector 0 by Theorem 6.1.3.
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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 Rn.
To show that it is a subspace of Rn we must show that it is closed under scalar multiplication
Let u and v be any vectors in ker(T), and let c be any scalar.
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6.3. Kernel and Range
Kernel of A Matrix Transformation
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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{v1, v2, v3}
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6.3. Kernel and Range
Kernel of A Matrix Transformation
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,
and
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
which shows that cu and u+v are images of vectors in S.
Thus, W is closed under scalar multiplication and addition.
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6.3. Kernel and Range
Range of A Linear Transformation
The range of a linear transformation T:RnRm 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.
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6.3. Kernel and Range
Range of A Matrix Transformation
If TA:RnRm is the linear transformation corresponding to the matrix A, then the range of TA
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.
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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
Rm is in the range of a linear transformation T:RnRm. 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.
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6.3. Kernel and Range
Existence and Uniqueness Issues
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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.
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6.3. Kernel and Range
Existence and Uniqueness Issues
(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).
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6.3. Kernel and Range
Existence and Uniqueness Issues
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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 Rn.
Combining this with Theorem 6.3.12 and 6.3.13 completes the proof.
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6.3. Kernel and Range
A Unifying Theorem
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6.3. Kernel and Range
A Unifying Theorem
Example 13
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.
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.
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6.4. 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, “T2
circle T1”)
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6.4. Composition and Invertibility
Compositions of Linear Transformations
Let u and v be any vectors in Rn, and let c be a scalar.
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6.4. Composition and Invertibility
Compositions of Linear Transformations
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6.4. Composition and Invertibility
Compositions of Linear Transformations
Example 1
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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 2θ2-2θ1.
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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.
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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.
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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 (T3◦T2◦T1):RnRm by
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6.4. Composition and Invertibility
Composition of Three or More Linear Transformations
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.
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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.
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6.4. Composition and Invertibility
Composition of Three or More Linear Transformations
Example 5
Suppose that a vector in R3 is first rotated 45&ordm; about the positive x-axis, then the resulting
vector is rotated 45&ordm; about the positive y-axis, and then that vector is rotated 45&ordm; about the
positive z-axis. Find an appropriate axis and angle of rotation that achieves the same result in
one rotation.
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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 e1.
(17)
Also, it follows from Formula (16) of Section 6.2 that the angle of rotation satisfies
(16)
from which it follows that θ ≈ 64.74&ordm;
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6.4. Composition and Invertibility
Factoring Linear Operators into Compositions
Example 6
A diagonal matrix
can be factored as
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6.4. Composition and Invertibility
Factoring Linear Operators into Compositions
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.
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6.4. Composition and Invertibility
Factoring Linear Operators into Compositions
Example 7
The 2&times;2 elementary matrices have five possible forms:
If k&lt;0, then we can express k in the form k=-k1, where k1&gt;0, and we can factor the type 4 and 5
matrices as
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6.4. Composition and Invertibility
Factoring Linear Operators into Compositions
Recall from Theorem 3.3.3 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.
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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: RnRm
(13)
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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 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.
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6.4. Composition and Invertibility
Invertible Linear Operators
Example 9
Example 11
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6.4. Composition and Invertibility
Geometric Properties of Invertible Linear Operators on R2
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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 6.4.7.
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6.5. Computer Graphics
Wireframes, Matrix Representations of Wireframes
Example 1
wire
wireframe
vertices
connectivity matrix
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6.5. Computer Graphics
Transforming Wireframes
Example 3
Shearing the roman version in the position x-direction to an angle that is 15&ordm; off the vertical.
The connectivity matrix does not change.
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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=(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.
Example 4
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6.5. Computer Graphics
Translation Using Homogeneous Coordinates
Example 5
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6.5. Computer Graphics
Translation Using Homogeneous Coordinates
Example 6
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6.5. Computer Graphics
Translation Using Homogeneous Coordinates
Example 7
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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 zaxis, 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.
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6.5. Computer Graphics
Three-Dimensional Graphics
vanishing point
perspective projection
orthogonal projection
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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.
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6.5. Computer Graphics
Three-Dimensional Graphics
Example 8
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6.5. Computer Graphics
Three-Dimensional Graphics
Example 8
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