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CHAPTER 6�LINEAR TRANSFORMATIONS

Elementary Linear Algebra

R. Larson (8 Edition)

6.1 Introduction to Linear Transformations

6.2 The Kernel and Range of a Linear Transformation

6.3 Matrices for Linear Transformations

6.4 Transition Matrices and Similarity

6.5 Applications of Linear Transformations

投影片設計製作者

淡江大學電機系翁慶昌教授

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CH 6 Linear Algebra Applied

Multivariate Statistics (p.304) Circuit Design (p.322)

Control Systems (p.314)

Population Age and Growth Distribution (p.331) Computer Graphics (p.338)

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6.1 Introduction to Linear Transformations

Function T that maps a vector space V into a vector space W:

V: the domain of T

W: the codomain of T

Elementary Linear Algebra: Section 6.1, p.298

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Image of v under T:

If v is in V and w is in W such that

Then w is called the image of v under T .

the range of T:

The set of all images of vectors in V.

the preimage of w:

The set of all v in V such that T(v)=w.

Elementary Linear Algebra: Section 6.1, p.298

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Ex 1: (A function from R²into R² )

(a) Find the image of v=(-1,2). (b) Find the preimage of w=(-1,11)

Sol:

Thus {(3, 4)} is the preimage of w=(-1, 11).

Elementary Linear Algebra: Section 6.1, p.298

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Linear Transformation (L.T.):

Elementary Linear Algebra: Section 6.1, p.299

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Notes:

(1) A linear transformation is said to be operation preserving.

Addition in V

Addition in W

Scalar multiplication in V

Scalar multiplication in W

(2) A linear transformation from a vector space into itself is called a linear operator.

Elementary Linear Algebra: Section 6.1, p.299

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Ex 2: (Verifying a linear transformation T from R² into R²)

Pf:

Elementary Linear Algebra: Section 6.1, p.299

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Therefore, T is a linear transformation.

Elementary Linear Algebra: Section 6.1, p.299

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Ex 3: (Functions that are not linear transformations)

Elementary Linear Algebra: Section 6.1, p.300

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Notes: Two uses of the term “linear”.

(1) is called a linear function because its graph is a line.

(2) is not a linear transformation from a vector space R into R because it preserves neither vector addition nor scalar multiplication.

Elementary Linear Algebra: Section 6.1, p.300

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Zero transformation:

Identity transformation:

Thm 6.1: (Properties of linear transformations)

Elementary Linear Algebra: Section 6.1, p.300

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Ex 4: (Linear transformations and bases)

Let be a linear transformation such that

Sol:

(T is a L.T.)

Find T(2, 3, −2).

Elementary Linear Algebra: Section 6.1, p.301

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Ex 5: (A linear transformation defined by a matrix)

The function is defined as

Sol:

(vector addition)

(scalar multiplication)

Elementary Linear Algebra: Section 6.1, p.301

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Thm 6.2: (The linear transformation given by a matrix)

Let A be an m×n matrix. The function T defined by

is a linear transformation from Rⁿinto R^m.

Note:

Elementary Linear Algebra: Section 6.1, p.302

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Ex 7: (Rotation in the plane)

Show that the L.T. given by the matrix

has the property that it rotates every vector in R²counterclockwise about the origin through the angle θ.

Sol:

(polar coordinates)

r： the length of v

α：the angle from the positive x-axis counterclockwise to the vector v

Elementary Linear Algebra: Section 6.1, p.303

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r：the length of T(v)

θ +α：the angle from the positive x-axis counterclockwise to

the vector T(v)

Thus, T(v) is the vector that results from rotating the vector v counterclockwise through the angle θ.

Elementary Linear Algebra: Section 6.1, p.303

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Ex 8: (A projection in R³)

is called a projection in R³.

The linear transformation is given by

Elementary Linear Algebra: Section 6.1, p.304

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Ex 9: (A linear transformation from M_m_×_ninto M_n_×_m)

Show that T is a linear transformation.

Sol:

Therefore, T is a linear transformation from M_m_×_ninto M_n_×_m.

Elementary Linear Algebra: Section 6.1, p.304

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Key Learning in Section 6.1

Find the image and preimage of a function.
Show that a function is a linear transformation, and find a linear transformation.

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Keywords in Section 6.1

function: 函數
domain: 論域
codomain: 對應論域
image of v under T: 在T映射下v的像
range of T: T的值域
preimage of w: w的反像
linear transformation: 線性轉換
linear operator: 線性運算子
zero transformation: 零轉換
identity transformation: 相等轉換

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6.2 The Kernel and Range of a Linear Transformation

Kernel of a linear transformation T:

Let be a linear transformation

Then the set of all vectors v in V that satisfy is called the kernel of T and is denoted by ker(T).

Ex 1: (Finding the kernel of a linear transformation)

Sol:

Elementary Linear Algebra: Section 6.2, p.309

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Ex 2: (The kernel of the zero and identity transformations)

(a) T(v) = 0 (the zero transformation )

(b) T(v) = v (the identity transformation )

Ex 3: (Finding the kernel of a linear transformation)

Sol:

Elementary Linear Algebra: Section 6.2, p.309

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Ex 5: (Finding the kernel of a linear transformation)

Sol:

Elementary Linear Algebra: Section 6.2, p.310

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Elementary Linear Algebra: Section 6.2, p.310

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Thm 6.3: (The kernel is a subspace of V)

The kernel of a linear transformation is a subspace of the domain V.

Pf:

Note:

The kernel of T is sometimes called the nullspace of T.

Elementary Linear Algebra: Section 6.2, p.311

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Ex 6: (Finding a basis for the kernel)

Find a basis for ker(T) as a subspace of R⁵.

Elementary Linear Algebra: Section 6.2, p.311

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Sol:

Elementary Linear Algebra: Section 6.2, p.311

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Corollary to Thm 6.3:

Range of a linear transformation T:

Elementary Linear Algebra: Section 6.2, p.311-312

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Thm 6.4: (The range of T is a subspace of W)

Pf:

Elementary Linear Algebra: Section 6.2, p.312

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Notes:

Corollary to Thm 6.4:

Elementary Linear Algebra: Section 6.2, p.312

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Ex 7: (Finding a basis for the range of a linear transformation)

Find a basis for the range of T.

Elementary Linear Algebra: Section 6.2, p.313

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Sol:

Elementary Linear Algebra: Section 6.2, p.313

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Rank of a linear transformation T:V→W:

Nullity of a linear transformation T:V→W:

Note:

Elementary Linear Algebra: Section 6.2, p.313

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Thm 6.5: (Sum of rank and nullity)

Pf:

Elementary Linear Algebra: Section 6.2, p.313

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Ex 8: (Finding the rank and nullity of a linear transformation)

Sol:

Elementary Linear Algebra: Section 6.2, p.314

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Ex 9: (Finding the rank and nullity of a linear transformation)

Sol:

Elementary Linear Algebra: Section 6.2, p.314

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One-to-one:

Elementary Linear Algebra: Section 6.2, p.315

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Onto:

(T is onto W when W is equal to the range of T.)

Elementary Linear Algebra: Section 6.2, p.315

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Thm 6.6: (One-to-one linear transformation)

Pf:

Elementary Linear Algebra: Section 6.2, p.315

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Ex 10: (One-to-one and not one-to-one linear transformation)

Elementary Linear Algebra: Section 6.2, p.315

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Thm 6.7: (Onto linear transformation)

Thm 6.8: (One-to-one and onto linear transformation)

Pf:

Elementary Linear Algebra: Section 6.2, p.316

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Ex 11:

Sol:

T:Rⁿ→R^m	dim(domain of T)	rank(T)	nullity(T)	1-1	onto
(a)T:R³→R³	3	3	0	Yes	Yes
(b)T:R²→R³	2	2	0	Yes	No
(c)T:R³→R²	3	2	1	No	Yes
(d)T:R³→R³	3	2	1	No	No

Elementary Linear Algebra: Section 6.2, p.316

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Isomorphism:

Thm 6.9: (Isomorphic spaces and dimension)

Pf:

Two finite-dimensional vector space V and W are isomorphic if and only if they are of the same dimension.

Elementary Linear Algebra: Section 6.2, p.317

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It can be shown that this L.T. is both 1-1 and onto.

Thus V and W are isomorphic.

Elementary Linear Algebra: Section 6.2, p.317

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Ex 12: (Isomorphic vector spaces)

The following vector spaces are isomorphic to each other.

Elementary Linear Algebra: Section 6.2, p.317

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Key Learning in Section 6.2

Find the kernel of a linear transformation.
Find a basis for the range, the rank, and the nullity of a linear transformation.
Determine whether a linear transformation is one-to-one or onto.
Determine whether vector spaces are isomorphic.

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Keywords in Section 6.2

kernel of a linear transformation T: 線性轉換T的核空間
range of a linear transformation T: 線性轉換T的值域
rank of a linear transformation T: 線性轉換T的秩
nullity of a linear transformation T: 線性轉換T的核次數
one-to-one: 一對一
onto: 映成
isomorphism(one-to-one and onto): 同構
isomorphic space: 同構的空間

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6.3 Matrices for Linear Transformations

Three reasons for matrix representation of a linear transformation:

It is simpler to write.
It is simpler to read.
It is more easily adapted for computer use.

Two representations of the linear transformation T:R³→R³:

Elementary Linear Algebra: Section 6.3, p.320

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Thm 6.10: (Standard matrix for a linear transformation)

Elementary Linear Algebra: Section 6.3, p.320

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Pf:

Elementary Linear Algebra: Section 6.3, p.321

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Elementary Linear Algebra: Section 6.3, p.321

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Ex 1: (Finding the standard matrix of a linear transformation)

Sol:

Vector Notation Matrix Notation

Elementary Linear Algebra: Section 6.3, p.321

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Note:

Check:

Elementary Linear Algebra: Section 6.3, p.321

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Ex 2: (Finding the standard matrix of a linear transformation)

Sol:

Notes:

(1) The standard matrix for the zero transformation from Rⁿinto R^m

is the m × n zero matrix.

(2) The standard matrix for the zero transformation from Rⁿinto Rⁿ

is the n × n identity matrix I_n.

Elementary Linear Algebra: Section 6.3, p.322

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Composition of T₁:Rⁿ→R^mwith T₂:R^m→R^p:

Thm 6.11: (Composition of linear transformations)

Elementary Linear Algebra: Section 6.3, p.323

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Pf:

Note:

Elementary Linear Algebra: Section 6.3, p.323

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Ex 3: (The standard matrix of a composition)

Sol:

Elementary Linear Algebra: Section 6.3, p.324

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Elementary Linear Algebra: Section 6.3, p.324

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Inverse linear transformation:

Note:

If the transformation T is invertible, then the inverse is unique and denoted by T^–1.

Elementary Linear Algebra: Section 6.3, p.324

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Thm 6.12: (Existence of an inverse transformation)

Note:

If T is invertible with standard matrix A, then the standard matrix for T^–1 is A^–1.

T is invertible.
T is an isomorphism.
A is invertible.

Elementary Linear Algebra: Section 6.3, p.325

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Ex 4: (Finding the inverse of a linear transformation)

Sol:

Show that T is invertible, and find its inverse.

Elementary Linear Algebra: Section 6.3, p.325

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Elementary Linear Algebra: Section 6.3, p.325

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the matrix of T relative to the bases B and B':

Thus, the matrix of T relative to the bases B and B' is

Elementary Linear Algebra: Section 6.3, p.326

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Transformation matrix for nonstandard bases:

Elementary Linear Algebra: Section 6.3, p.326

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Elementary Linear Algebra: Section 6.3, p.326

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Ex 5: (Finding a matrix relative to nonstandard bases)

Sol:

Elementary Linear Algebra: Section 6.3, p.326

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Ex 6:

Sol:

Check:

Elementary Linear Algebra: Section 6.3, p.327

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Notes:

Elementary Linear Algebra, Section 6.3, p.327

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Key Learning in Section 6.3

Find the standard matrix for a linear transformation.
Find the standard matrix for the composition of linear transformations and find the inverse of an invertible linear transformation.
Find the matrix for a linear transformation relative to a nonstandard basis.

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Keywords in Section 6.3

standard matrix for T: T 的標準矩陣
composition of linear transformations: 線性轉換的合成
inverse linear transformation: 反線性轉換
matrix of T relative to the bases B and B' : T對應於基底B到B'的矩陣
matrix of T relative to the basis B: T對應於基底B的矩陣

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6.4 Transition Matrices and Similarity

Elementary Linear Algebra: Section 6.4, p.330

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Two ways to get from to :

Elementary Linear Algebra: Section 6.4, p.330

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Ex 1: (Finding a matrix for a linear transformation)

Sol:

Elementary Linear Algebra: Section 6.4, p.330

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Elementary Linear Algebra: Section 6.4, p.330

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Ex 2: (Finding a matrix for a linear transformation)

Sol:

Elementary Linear Algebra: Section 6.4, p.331

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Ex 3: (Finding a matrix for a linear transformation)

Sol:

Elementary Linear Algebra: Section 6.4, p.331

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Similar matrix:

For square matrices A and A‘ of order n, A‘ is said to be similar to A if there exist an invertible matrix P s.t.

Thm 6.13: (Properties of similar matrices)

Let A, B, and C be square matrices of order n.

Then the following properties are true.

(1) A is similar to A.

(2) If A is similar to B, then B is similar to A.

(3) If A is similar to B and B is similar to C, then A is similar to C.

Pf:

Elementary Linear Algebra: Section 6.4, p.332

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Ex 4: (Similar matrices)

Elementary Linear Algebra: Section 6.4, p.332

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Ex 5: (A comparison of two matrices for a linear transformation)

Sol:

Elementary Linear Algebra: Section 6.4, p.333

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Elementary Linear Algebra: Section 6.4, p.333

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Notes: Computational advantages of diagonal matrices:

Elementary Linear Algebra: Section 6.4, p.333

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Key Learning in Section 6.4

Find and use a matrix for a linear transformation.
Show that two matrices are similar and use the properties of similar matrices.

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Keywords in Section 6.4

matrix of T relative to B: T 相對於B的矩陣
matrix of T relative to B' : T 相對於B'的矩陣
transition matrix from B' to B : 從B'到B的轉移矩陣
transition matrix from B to B' : 從B到B'的轉移矩陣
similar matrix: 相似矩陣

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6.5 Applications of Linear Transformations

Elementary matrices for linear transformations in R²:

Reflection in y-axis

Reflection in x-axis

Reflection in line y = x

Horizontal expansion (k > 1)

or contraction (0 < k < 1)

Vertical expansion (k > 1)

or contraction (0 < k < 1)

Horizontal shear

Vertical shear

Elementary Linear Algebra: Section 6.5, p.336

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Ex 1: (Reflections in R²)

a. Reflection

in the y-axis

b. Reflection

in the x-axis

c. Reflection

in the line y = x

Elementary Linear Algebra: Section 6.5, p.336

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Ex 2: (Expansions and Contractions in R²)

a. Horizontal contractions

and expansions

b. Vertical contractions

and expansions

Elementary Linear Algebra: Section 6.5, p.337

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Ex 3: (Shears in R²)

a. Horizontal shear

Under this transformation, points in the upper half-plane “shear” to the right by amounts proportional to their y-coordinates. Points in the lower half-plane “shear” to the left by amounts proportional to the absolute values of their y-coordinates. Points on the x-axis do not move by this transformation.

Elementary Linear Algebra: Section 6.5, p.338

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b. Vertical shear

Here, points in the right half-plane “shear” upward by amounts proportional to their x-coordinates. Points in the left half-plane “shear” downward by amounts proportional to the absolute values of their x-coordinates. Points on the y-axis do not move.

Elementary Linear Algebra: Section 6.5, p.338

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Rotation in R³:

Suppose you want to rotate the point (x, y, z) counterclockwise about the z-axis through an angle θ.

Elementary Linear Algebra: Section 6.5, p.338

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Ex 4: (Rotation About the z-axis)

Sol:

a. A rotation of

Elementary Linear Algebra: Section 6.5, p.339

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b. A rotation of

c. A rotation of

Elementary Linear Algebra: Section 6.5, p.339

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Rotation about the z-axis

Rotation about the y-axis

Rotation about the x-axis

Elementary Linear Algebra: Section 6.5, p.340

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Note:

To illustrate the right-hand rule, imagine the thumb of your right hand pointing in the positive direction of an axis. The cupped fingers will point in the direction of counterclockwise rotation. The figure below shows counterclockwise rotation about the z-axis.

Elementary Linear Algebra: Section 6.5, p.340

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Ex 5: (Rotation About the x-Axis and y-Axis)

(a) A rotation of about the x-axis

(b) A rotation of about the y-axis

Elementary Linear Algebra: Section 6.5, p.340

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Key Learning in Section 6.5

Identify linear transformations defined by reflections, expansions, contractions, or shears in R².
Use a linear transformation to rotate a figure in R³.

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Keywords in Section 6.5

reflection: 鏡射
expansion: 擴展
contraction: 縮減
share: 切變
rectangular prism: 長方體

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Multivariate Statistics

Many multivariate statistical methods can use linear transformations. For instance, in a multiple regression analysis, there are two or more independent variables and a single dependent variable. A linear transformation is useful for finding weights to be assigned to the independent variables to predict the value of the dependent variable. Also, in a canonical correlation analysis, there are two or more independent variables and two or more dependent variables. Linear transformations can help find a linear combination of the independent variables to predict the value of a linear combination of the dependent variables.

6.1 Linear Algebra Applied

Elementary Linear Algebra: Section 6.1, p.304

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Control Systems

A control system, such as the one shown for a dairy factory, processes an input signal x_k and produces an output signal x_k₊₁.Without external feedback, the difference equation x_k₊₁= Ax_k, a linear transformation where x_iis an n × 1 vector and A is an n × n matrix, can model the relationship between the input and output signals. Typically, however, a control system has external feedback, so the relationship becomes x_k₊₁= Ax_k+Bu_k, where B is an n 🞨m matrix and u_kis an m × 1 input, or control, vector. A system is called controllable when it can reach any desired final state from its initial state in or fewer steps. If A and B make up a controllable system, then the rank of the controllability matrix

[B AB A²B . . . Aⁿ^-1B]

is equal to n.

6.2 Linear Algebra Applied

Elementary Linear Algebra: Section 6.2, 314

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Circuit Design

Ladder networks are useful tools for electrical engineers involved in circuit design. In a ladder network, the output voltage V and current I of one circuit are the input voltage and current of the circuit next to it. In the ladder network shown below, linear transformations can relate the input and output of an individual circuit (enclosed in a dashed box). Using Kirchhoff’s Voltage and Current Laws and Ohm’s Law,

and

6.3 Linear Algebra Applied

Elementary Linear Algebra: Section 6.3, p.322

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Circuit Design

A composition can relate the input and output of the entire ladder network, that is, V₁ and I₁ to V₃ and I₃. Discussion on the composition of linear transformations begins on the following page.

6.3 Linear Algebra Applied

Elementary Linear Algebra: Section 6.3, p.322

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Weather

A Leslie matrix, named after British mathematician Patrick H. Leslie (1900–1974), can be used to find the age and growth distributions of a population over time. The entries in the first row of an n × n Leslie matrix L are the average numbers of offspring per member for each of n age classes. The entries in subsequent rows are pi in row i +1, column i and 0 elsewhere, where pi is the probability that an ith age class member will survive to become an (i +1)th age class member. If x_jis the age distribution vector for the jth time period, then the age distribution vector for the (j +1)th time period can be found using the linear transformation x_j₊₁ = Lx_j. You will study population growth models using Leslie matrices in more detail in Section 7.4.

6.4 Linear Algebra Applied

Elementary Linear Algebra: Section 6.4, p.331

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Computer Graphics

The use of computer graphics is common in many fields. By using graphics software, a designer can “see” an object before it is physically created. Linear transformations can be useful in computer graphics. To illustrate with a simplified example, only 23 points in R³were used to generate images of the toy boat shown in the figure at the left. Most graphics software can use such minimal information to generate views of an image from any perspective, as well as color, shade, and render as appropriate. Linear transformations, specifically those that produce rotations in R³can represent the different views. The remainder of this section discusses rotation in R³.

6.5 Linear Algebra Applied

Elementary Linear Algebra: Section 6.5, p.338

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