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Elementary Linear Algebra
Anton & Rorres, 9th Edition
Lecture Set – 04
Chapter 4:
Euclidean Vector Spaces
Chapter Content



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Euclidean n-Space
Linear Transformations from Rn to Rm
Properties of Linear Transformations Rn to Rm
Linear Transformations and Polynomials
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Elementary Linear Algebra
2
4-1 Definitions

If n is a positive integer, then an ordered n-tuple is a sequence of n
real numbers (a1,a2,…,an). The set of all ordered n-tuple is called nspace and is denoted by Rn.

Two vectors u = (u1 ,u2 ,…,un) and v = (v1 ,v2 ,…, vn) in Rn are called
equal if
u1 = v1 ,u2 = v2 , …, un = vn
The sum u + v is defined by
u + v = (u1+v1 , u1+v1 , …, un+vn)
and if k is any scalar, the scalar multiple ku is defined by
ku = (ku1 ,ku2 ,…,kun)
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4-1 Remarks

The operations of addition and scalar multiplication in
this definition are called the standard operations on Rn.

The zero vector in Rn is denoted by 0 and is defined to be
the vector 0 = (0, 0, …, 0).
If u = (u1 ,u2 ,…,un) is any vector in Rn, then the negative
(or additive inverse) of u is denoted by -u and is defined
by -u = (-u1 ,-u2 ,…,-un).


The difference of vectors in Rn is defined by
v – u = v + (-u) = (v1 – u1 ,v2 – u2 ,…,vn – un)
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Theorem 4.1.1 (Properties of Vector in Rn)

If u = (u1 ,u2 ,…,un), v = (v1 ,v2 ,…, vn), and w = (w1 ,w2 ,…,
wn) are vectors in Rn and k and l are scalars, then:




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u+v=v+u
u + (v + w) = (u + v) + w
u+0=0+u=u
u + (-u) = 0; that is u – u = 0
k(lu) = (kl)u
k(u + v) = ku + kv
(k+l)u = ku+lu
1u = u
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4-1 Euclidean Inner Product

Definition


If u = (u1 ,u2 ,…,un), v = (v1 ,v2 ,…, vn) are vectors in Rn, then the
Euclidean inner product u · v is defined by
u · v = u1 v1 + u2 v2 + … + un vn
Example 1

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The Euclidean inner product of the vectors u = (-1,3,5,7) and v =
(5,-4,7,0) in R4 is
u · v = (-1)(5) + (3)(-4) + (5)(7) + (7)(0) = 18
Elementary Linear Algebra
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Theorem 4.1.2
Properties of Euclidean Inner Product

If u, v and w are vectors in Rn and k is any scalar, then
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u·v=v·u
(u + v) · w = u · w + v · w
(k u) · v = k(u · v)
v · v ≥ 0; Further, v · v = 0 if and only if v = 0
Elementary Linear Algebra
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4-1 Example 2

(3u + 2v) · (4u + v)
= (3u) · (4u + v) + (2v) · (4u + v )
= (3u) · (4u) + (3u) · v + (2v) · (4u) + (2v) · v
=12(u · u) + 11(u · v) + 2(v · v)
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4-1 Norm and Distance in Euclidean n-Space

We define the Euclidean norm (or Euclidean length) of a
vector u = (u1 ,u2 ,…,un) in Rn by
u  (u  u)1/ 2  u12  u22  ...  un2

Similarly, the Euclidean distance between the points u =
(u1 ,u2 ,…,un) and v = (v1 , v2 ,…,vn) in Rn is defined by
d (u, v)  u  v  (u1  v1 ) 2  (u2  v2 ) 2  ...  (un  vn ) 2
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4-1 Example 3

Example

If u = (1,3,-2,7) and v = (0,7,2,2), then in the Euclidean space R4
u  (1) 2  (3) 2  (2) 2  (7) 2  63  3 7
d (u, v )  (1  0) 2  (3  7) 2  (2  2) 2  (7  2) 2  58
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Theorem 4.1.3
(Cauchy-Schwarz Inequality in Rn)

If u = (u1 ,u2 ,…,un) and v = (v1 , v2 ,…,vn) are vectors in Rn,
then |u · v| ≤ || u || || v ||
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Theorem 4.1.4
(Properties of Length in Rn)

If u and v are vectors in Rn and k is any scalar, then

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|| u || ≥ 0
|| u || = 0 if and only if u = 0
|| ku || = | k ||| u ||
|| u + v || ≤ || u || + || v ||
(Triangle inequality)
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Theorem 4.1.5
(Properties of Distance in Rn)

If u, v, and w are vectors in Rn and k is any scalar, then

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d(u, v) ≥ 0
d(u, v) = 0 if and only if u = v
d(u, v) = d(v, u)
d(u, v) ≤ d(u, w ) + d(w, v)
(Triangle inequality)
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Theorem 4.1.6

If u, v, and w are vectors in Rn with the Euclidean inner
product, then u · v = ¼ || u + v ||2–¼ || u–v ||2
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4-1 Orthogonality

Two vectors u and v in Rn are called orthogonal if u · v = 0

Example 4

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In the Euclidean space R4 the vectors
u = (-2, 3, 1, 4) and v = (1, 2, 0, -1)
are orthogonal, since
u · v = (-2)(1) + (3)(2) + (1)(0) + (4)(-1) = 0
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Theorem 4.1.7
(Pythagorean Theorem in Rn)

If u and v are orthogonal vectors in Rn which the
Euclidean inner product, then || u + v ||2 = || u ||2 + || v ||2
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4-1 Matrix Formulae for the Dot Product

If we use column matrix notation for the vectors
u = [u1 u2 … un]T and v = [v1 v2 … vn]T ,
or
 u1 
u     and
un 
 v1 
v    
vn 
then
u · v = v Tu
Au · v = u · ATv
u · Av = ATu · v
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4-1 Example 5

Verifying that Au‧v= u‧Atv
 1  2 3
 1
  2
A   2
4 1, u   2 , v   0 
 1 0 1
 4 
 5 
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4-1 A Dot Product View of
Matrix Multiplication


If A = [aij] is an mr matrix and B =[bij] is an rn matrix, then the ijthe entry of AB is
ai1b1j + ai2b2j + ai3b3j + … + airbrj
which is the dot product of the ith row vector of A and the jth
column vector of B
Thus, if the row vectors of A are r1, r2, …, rm and the column
vectors of B are c1, c2, …, cn ,
r1  c1 r1  c 2  r1  c n
r  c r  c  r  c
2
n
AB   2 1 2 2
 



rm  c1 rm  c 2  rm  c n
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Elementary Linear Algebra






19
4-1 Example 6

A linear system written in dot product form
system
3x1  4 x2  x3  1
2 x1  7 x2  4 x3  5
x1  5 x2  8 x3  0
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dot product form
 (3,-4,1)  ( x1 , x2 , x3 )  1
(2,-7,-4) ( x , x , x )  5
1
2
3 

 
 (1,5,-8)  ( x1 , x2 , x3 )  0
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Chapter Content




Euclidean n-Space
Linear Transformations from Rn to Rm
Properties of Linear Transformations Rn to Rm
Linear Transformations and Polynomials
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Elementary Linear Algebra
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4-2 Functions from
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
n
R
to R
A function is a rule f that associates with each element in
a set A one and only one element in a set B.
If f associates the element b with the element, then we
write b = f(a) and say that b is the image of a under f or
that f(a) is the value of f at a.
The set A is called the domain of f and the set B is called
the codomain of f.
The subset of B consisting of all possible values for f as a
varies over A is called the range of f.
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4-2 Examples
Formula
f (x )
f ( x, y )
f ( x, y , z )
f ( x1 , x2 ,..., xn )
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Example
f ( x)  x 2
f ( x, y)  x 2  y 2
f ( x, y, z )  x 2
Classification
Real-valued function of a Function from
real variable
R to R
Real-valued function of
two real variable
Function from
R2 to R
Real-valued function of
three real variable
Function from
R3 to R
Real-valued function of
n real variable
Function from
Rn to R
 y2  z2
f ( x1 , x2 ,..., xn ) 
x12  x22  ...  xn2
Description
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4-2 Function from

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n
R
to
m
R
If the domain of a function f is Rn and the codomain is Rm, then f is
called a map or transformation from Rn to Rm. We say that the
function f maps Rn into Rm, and denoted by f : Rn  Rm.
If m = n the transformation f : Rn  Rm is called an operator on Rn.
Suppose f1, f2, …, fm are real-valued functions of n real variables,
say
w1 = f1(x1,x2,…,xn)
w2 = f2(x1,x2,…,xn)
…
wm = fm(x1,x2,…,xn)
These m equations define a transformation from Rn to Rm. If we
denote this transformation by T: Rn  Rm then
T (x1,x2,…,xn) = (w1,w2,…,wm)
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4-2 Example 1

The equations
w1 = x1 + x2
w2 = 3x1x2
w3 = x12 – x22
define a transformation T: R2  R3.
T(x1, x2) = (x1 + x2, 3x1x2, x12 – x22)
Thus, for example, T(1,-2) = (-1,-6,-3).
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4-2 Linear Transformations
from Rn to Rm

A linear transformation (or a linear operator if m = n) T: Rn  Rm is
defined by equations of the form
w1  a11x1  a12 x2  ...  a1n xn
w2  a21x1  a22 x2  ...  a2 n xn
or




wm  am1 x1  am 2 x2  ...  amn xn
 w1  a11 a12 
 w  a a 
 2    21 22
   

  
 wm  amn amn 
a13   x1 
a23   x2 
   
 
amn   xm 
or
w = Ax

The matrix A = [aij] is called the standard matrix for the linear
transformation T, and T is called multiplication by A.
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4-2 Example 2 (Linear Transformation)

The linear transformation T : R4  R3 defined by the
equations
w1 = 2x1 – 3x2 + x3 – 5x4
w2 = 4x1 + x2 – 2x3 + x4
w3 = 5x1 – x2 + 4x3
the standard matrix for T (i.e., w = Ax) is
2  3 1  5
A  4 1  2 1 
5  1 4 0 
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4-2 Notations of Linear Transformations

If it is important to emphasize that A is the standard matrix for
T. We denote the linear transformation T: Rn  Rm by TA: Rn 
Rm . Thus, TA(x) = Ax
 We can also denote the standard matrix for T by the symbol
[T], or T(x) = [T]x

Remark:

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A correspondence between mn matrices and linear transformations
from Rn to Rm :
 To each matrix A there corresponds a linear transformation TA
(multiplication by A), and to each linear transformation T: Rn  Rm,
there corresponds an mn matrix [T] (the standard matrix for T).
Elementary Linear Algebra
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4-2 Examples

Example 3 (Zero Transformation from Rn to Rm)



If 0 is the mn zero matrix and 0 is the zero vector in Rn, then for
every vector x in Rn
T0(x) = 0x = 0
So multiplication by zero maps every vector in Rn into the zero
vector in Rm. We call T0 the zero transformation from Rn to Rm.
Example 4 (Identity Operator on Rn)
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If I is the nn identity, then for every vector in Rn
TI(x) = Ix = x
So multiplication by I maps every vector in Rn into itself.
We call TI the identity operator on Rn.
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4-2 Reflection Operators

In general, operators on R2 and R3 that map each vector
into its symmetric image about some line or plane are
called reflection operators.

Such operators are linear.
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4-2 Reflection Operators (2-Space)
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4-2 Reflection Operators (3-Space)
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4-2 Projection Operators

In general, a projection operator (or more precisely an
orthogonal projection operator) on R2 or R3 is any
operator that maps each vector into its orthogonal
projection on a line or plane through the origin.

The projection operators are linear.
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4-2 Projection Operators
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4-2 Projection Operators
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4-2 Rotation Operators

An operator that rotate each vector in R2 through a
fixed angle  is called a rotation operator on R2.
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4-2 Example 6

If each vector in R2 is rotated through an angle of /6
(30) ,then the image w of a vector
x 
x 
 y
is

cos 
6
w
sin 
6

 3
 x
6    2
 y  


  1
cos
6 
 2
 sin 

 3 x  1 y
2 x    2
2 
 y 

3
3


1
y
 2 x 
2 
2 
1
For example, the image of the vector

3  1


1
2

x    is w  
1 
3
1


2


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4-2 A Rotation of Vectors in

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
3
R
A rotation of vectors in R3 is usually
described in relation to a ray emanating
from the origin, called the axis of rotation.
As a vector revolves around the axis of
rotation it sweeps out some portion of a
cone.
The angle of rotation is described as
“clockwise” or “counterclockwise” in
relation to a viewpoint that is along the axis
of rotation looking toward the origin.
The axis of rotation can be specified by a
nonzero vector u that runs along the axis of
rotation and has its initial point at the origin.
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4-2 A Rotation of Vectors in
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3
R
39
4-2 Dilation and Contraction Operators

If k is a nonnegative scalar, the operator on R2 or R3 is
called a contraction with factor k if 0 ≤ k ≤ 1 and a
dilation with factor k if k ≥ 1 .
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4-2 Compositions of Linear Transformations

If TA : Rn  Rk and TB : Rk  Rm are linear transformations, then for
each x in Rn one can first compute TA(x), which is a vector in Rk, and
then one can compute TB(TA(x)), which is a vector in Rm.


the application of TA followed by TB produces a transformation from Rn
to Rm.
This transformation is called the composition of TB with TA and is
denoted by TB。TA. Thus (TB 。 TA)(x) = TB(TA(x))

The composition TB 。 TA is linear since
(TB 。TA)(x) = TB(TA(x)) = B(Ax) = (BA)x

The standard matrix for TB。TA is BA. That is, TB。TA = TBA
Multiplying matrices is equivalent to composing the corresponding
linear transformations in the right-to-left order of the factors.

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4-2 Example 6
(Composition of Two Rotations)



Let T1 : R2  R2 and T2 : R2 
R2 be linear operators that rotate
vectors through the angle 1 and
2, respectively.
The operation
(T2 。T1)(x) = (T2(T1(x)))
first rotates x through the angle
1, then rotates T1(x) through the
angle 2.
It follows that the net effect of
T2 。T1
is to rotate each vector in R2
through the angle 1+ 2
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4-2 Example 7
Composition Is Not Commutative
0 1 0 1  0 1 
T1  T2   [T1 ][T2 ]        
1 0 0 0 0 0
0 1  0 1 0 0
T2  T1   [T2 ][T1 ]        
0 0 1 0 1 0 
so T1  T2   T2  T1 
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4-2 Example 8

Let T1: R2  R2 be the reflection about the y-axis, and T2:
R2  R2 be the reflection about the x-axis.



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(T1◦T2)(x,y) = T1(x, -y) = (-x, -y)
(T2◦T1)(x,y) = T2(-x, y) = (-x, -y)
is called the reflection about the origin
加圖4.2.9
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4-2 Compositions of Three or More
Linear Transformations




Consider the linear transformations
T 1 : Rn  Rk , T2 : Rk  R l , T 3 : Rl  Rm
We can define the composition (T3◦T2◦T1) : Rn  Rm by
(T3◦T2◦T1)(x) : T3(T2(T1(x)))
This composition is a linear transformation and the standard matrix
for T3◦T2◦T1 is related to the standard matrices for T1,T2, and T3 by
[T3◦T2◦T1] = [T3][T2][T1]
If the standard matrices for T1, T2, and T3 are denoted by A, B, and C,
respectively, then we also have
TC◦TB◦TA = TCBA
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4-2 Example 9

Find the standard matrix for the linear operator T : R3  R3 that first
rotates a vector counterclockwise about the z-axis through an angle ,
then reflects the resulting vector about the yz-plane, and then
projects that vector orthogonally onto the xy-plane.
cos 
[T1 ]  sin 
 0
1 0
[T ]  0 1
0 0
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 sin 
cos 
0
0
0 ,
1
 1 0 0 
1 0 0 
[T2 ]   0 1 0, [T3 ]  0 1 0 
 0 0 1
0 0 0
0   1 0
0   0 1
0  0 0
0  cos 
0 sin 
1  0
 sin 
cos 
0
Elementary Linear Algebra
0  cos 
0    sin 
1 
0
sin 
cos 
0
0
0 
0 
46
Chapter Content




Euclidean n-Space
Linear Transformations from Rn to Rm
Properties of Linear Transformations Rn to Rm
Linear Transformations and Polynomials
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Elementary Linear Algebra
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4-3 One-to-One Linear Transformations

A linear transformation T : Rn →Rm is said to be one-toone if T maps distinct vectors (points) in Rn into distinct
vectors (points) in Rm

Remark:


That is, for each vector w in the range of a one-to-one linear
transformation T, there is exactly one vector x such that
T(x) = w.
Example 1


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Rotation operator is one-to-one
Orthogonal projection operator is not one-to-one
Elementary Linear Algebra
48
Theorem 4.3.1 (Equivalent Statements)

If A is an nn matrix and TA : Rn  Rn is multiplication by
A, then the following statements are equivalent.



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A is invertible
The range of TA is Rn
TA is one-to-one
Elementary Linear Algebra
49
4-3 Example 2 & 3

The rotation operator T : R2  R2 is one-to-one

The standard matrix for T is

[T] is invertible since
cos 
det
sin 

cos 
[T ]  
sin 
 sin  
cos  
 sin 
 cos 2   sin 2   1  0
cos 
The projection operator T : R3  R3 is not one-to-one

The standard matrix for T is

[T] is invertible since det[T] = 0
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1 0 0 
[T ]  0 1 0 
0 0 0
Elementary Linear Algebra
50
4-3 Inverse of a One-to-One Linear Operator




Suppose TA : Rn  Rn is a one-to-one linear operator
 The matrix A is invertible.
 TA-1 : Rn  Rn is itself a linear operator; it is called the inverse of TA.
 TA(TA-1(x)) = AA-1x = Ix = x and TA-1(TA (x)) = A-1Ax = Ix = x
 TA ◦ TA-1 = TAA-1 = TI and TA-1 ◦ TA = TA-1A = TI
If w is the image of x under TA, then TA-1 maps w back into x, since
TA-1(w) = TA-1(TA (x)) = x
When a one-to-one linear operator on Rn is written as T : Rn  Rn, then
the inverse of the operator T is denoted by T-1.
Thus, by the standard matrix, we have [T-1]=[T]-1
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Elementary Linear Algebra
51
4-3 Example 4

Let T : R2  R2 be the operator that rotates each vector in R2 through
the angle :
cos   sin  
[T ]  
sin 
cos  

Undo the effect of T means rotate each vector in R2 through the
angle -.

This is exactly what the operator T-1 does: the standard matrix T-1 is
sin   cos(  )  sin(  ) 
 cos 
1
1
[T ]  [T ]  




sin

cos

sin(


)
cos(


)

 

The only difference is that the angle  is replaced by -

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Elementary Linear Algebra
52
4-3 Example 5


Show that the linear operator T : R2  R2 defined by the equations
w1= 2x1+ x2
w2 = 3x1+ 4x2
is one-to-one, and find T-1(w1,w2).
Solution:
 4
w1  2 1   x1 
w   3 4  x 
 2 
 2 
 4
 5
w
1  1 
[T ]   
 w2   3
 5
T 1 ( w1 , w2 )  (
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2 1 
[T ]  

3 4
1
1 
 4
 
w

w2 
1

w


5
5
5
1
   

2   w2   3
2 
 w1  w2


5
5 
 5
 5
[T ]  [T ]  
 3
 5
1
1
1
 
5

2
5 
4
1
3
2
w1  w2 , w1  w2 )
5
5
5
5
Elementary Linear Algebra
53
Theorem 4.3.2
(Properties of Linear Transformations)

A transformation T : Rn  Rm is linear if and only if the following
relationships hold for all vectors u and v in Rn and every scalar c.
 T(u + v) = T(u) + T(v)
 T(cu) = cT(u)
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Elementary Linear Algebra
54
Theorem 4.3.3

If T : Rn  Rm is a linear transformation, and e1, e2, …, en are
the standard basis vectors for Rn, then the standard matrix for T
is
A = [T] = [T(e1) | T(e2) | … | T(en)]
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Elementary Linear Algebra
55
4-3 Example 6
(Standard Matrix for a Projection Operator)

Let l be the line in the xy-plane that passes through the
origin and makes an angle  with the positive x-axis,
where 0 ≤  ≤ . Let T: R2  R2 be a linear operator that
maps each vector into orthogonal projection on l.


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Find the standard matrix for T.
Find the orthogonal projection of
the vector x = (1,5) onto the line
through the origin that makes an
angle of  = /6 with the positive
x-axis.
Elementary Linear Algebra
56
4-3 Example 6 (continue)


The standard matrix for T can be written as
[T] = [T(e1) | T(e2)]
Consider the case 0    /2.

||T(e1)|| = cos 

 T (e1 ) cos    cos 2  
T (e1 )  


sin

cos

T
(
e
)
sin


 

1
||T(e2)|| = sin 
 T (e 2 )  cos   sin  cos  
T (e 2 )  


2
sin

T
(
e
)

sin





2

 cos 2  sin  cos  
T   

sin 2  
sin  cos 
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Elementary Linear Algebra
57
4-3 Example 6 (continue)

 cos 2  sin  cos  
T   

2
sin

cos

sin



Since sin (/6) = 1/2 and cos (/6) = 3 /2, it follows
from part (a) that the standard matrix for this projection
operator is
3 4
3 4
[T ]  

 3 4 1 4 
Thus,
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3  5 3 
 1   3 4
3 4 1   4 

T      
   
 5   3 4 1 4  5  3  5 


 4

Elementary Linear Algebra
58
4-3 Geometric Interpretation of Eigenvector


If T: Rn  Rn is a linear operator, then a scalar  is called an
eigenvalue of T if there is a nonzero x in Rn such that
T(x) = x
Those nonzero vectors x that satisfy this equation are called the
eigenvectors of T corresponding to 
Remarks:


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If A is the standard matrix for T, then the equation becomes
Ax = x
 The eigenvalues of T are precisely the eigenvalues of its standard
matrix A
 x is an eigenvector of T corresponding to  if and only if x is an
eigenvector of A corresponding to 
If  is an eigenvalue of A and x is a corresponding eigenvector, then Ax
= x, so multiplication by A maps x into a scalar multiple of itself
Elementary Linear Algebra
59
4-3 Example 7





Let T : R2  R2 be the linear operator that rotates each vector
through an angle .
If  is a multiple of , then every nonzero vector x is mapped onto
the same line as x, so every nonzero vector is an eigenvector of T.
cos   sin  
The standard matrix for T is A  
cos  
sin 
The eigenvalues of this matrix are the solutions of the characteristic
equation
  cos  sin 
det(I  A) 
0
 sin 
  cos 
That is, ( – cos )2 + sin2  = 0.
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Elementary Linear Algebra
60
4-3 Example 7(continue)
(  cos ) 2  sin 2   0



If  is not a multiple of 
 sin2 > 0
 no real solution for 
 A has no real eigenvectors.
If  is a multiple of 
 sin  = 0 and cos  = 1
In the case that sin  = 0 and
cos  = 1
  = 1 is the only eigenvalue

1 0
A

0
1


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


Thus, for all x in R2, T(x) = Ax
= Ix = x
So T maps every vector to
itself, and hence to the same
line.
In the case that sin  = 0 and
cos  = -1,
 A = -I and T(x) = -x
 T maps every vector to its
negative.
Elementary Linear Algebra
61
4-3 Example 8


Let T : R3  R3 be the orthogonal projection on xy-plane.
Vectors in the xy-plane



Every vector x along the z-axis



mapped into themselves under T
each nonzero vector in the xy-plane is an eigenvector corresponding to
the eigenvalue  = 1
mapped into 0 under T, which is on the same line as x
every nonzero vector on the z-axis is an eigenvector corresponding to
the eigenvalue 0
Vectors not in the xy-plane or along the z-axis


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mapped into  = 0 scalar multiples of themselves
there are no other eigenvectors or eigenvalues
Elementary Linear Algebra
62
4-3 Example 8 (continue)

The characteristic equation of A is
 1
det( I  A) 


0
0
0
 1
0
0
0  0 or (  1) 2   0

The eigenvectors of the matrix A corresponding to an eigenvalue λ
are the nonzero solutions of   1 0 0   x1  0
If  = 0, this system is
 1 0
 0 1

 0 0

1 0 0 
A  0 1 0 
0 0 0
 0

 0
 1
0
0  x1  0
0  x2   0
0   x3  0
0   x2   0
   x3  0
 x1  0
 x   0 
 2  
 x3  t 
The vectors are along the z-axis
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Elementary Linear Algebra
63
4-3 Example 8 (continue)

If  = 1, the system is
0 0 0  x1  0
0 0 0  x   0

 2   
0 0 1   x3  0

 x1   s 
 x   t 
 2  
 x3  0
The vectors are along the xy-plane
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Elementary Linear Algebra
64
Theorem 4.3.4 (Equivalent Statements)

If A is an nn matrix, and if TA : Rn  Rn is multiplication
by A, then the following are equivalent.









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A is invertible
Ax = 0 has only the trivial solution
The reduced row-echelon form of A is In
A is expressible as a product of elementary matrices
Ax = b is consistent for every n1 matrix b
Ax = b has exactly one solution for every n1 matrix b
det(A)  0
The range of TA is Rn
TA is one-to-one
Elementary Linear Algebra
65
Chapter Content




Euclidean n-Space
Linear Transformations from Rn to Rm
Properties of Linear Transformations Rn to Rm
Linear Transformations and Polynomials
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Elementary Linear Algebra
66
4-4 Example 1

Correspondence between polynomials and vectors


2017/5/6
Consider the quadratic function p(x)=ax2+bx+c
define the vector
a
 
z  b 
 c 
Elementary Linear Algebra
67
4-4 Example 2


Addition of polynomials by adding vectors
Let p(x)= 4x3-2x+1 and q(x)= 3x3-3x+x then to compute
r(x) = 4p(x)-2q(x)
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Elementary Linear Algebra
68
4-4 Example 3

Differentiation of polynomials

p(x) =ax2+bx+c
d
p( x)  2ax  b
dx
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Elementary Linear Algebra
69
4-4 Affine Transformation

An affine transformation from Rn to Rm is a mapping of
the form S(u) = T(u) + f, where T is a linear
transformation from Rn to Rm and f is a (constant) vector
in Rm.

Remark

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The affine transform S is a linear transformation if f is the
zero vector
Elementary Linear Algebra
70
4-4 Example 4 (Affine Transformations)

The mapping
 0 1
1
S (u)  
u


1
  1 0

transformation on R2.

is an affine
If u = (a,b), then
 0 1 a  1 b  1 
S (u)  
  




 1 0 b  1  a  1
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Elementary Linear Algebra
71
4-4 Interpolating Polynomials



Consider the problem of interpolating a polynomial to a
set of n+1 points (x0,y0), …, (xn,yn).
That is, we seek to find a curve p(x) = anxn + … + a0
The matrix
1 x0

1 x1
 

1 xn 1
1 x
n

x02

x12


xn21


xn2

x0n  a0   y0 
 y 
n 
a
x0   1   1 
      
 

n 
a
y
xn 1   n 1   n 1 
xnn  an   yn 
is known as a Vandermonde matrix
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Elementary Linear Algebra
72
4-4 Example 5 (Interpolating a Cubic)

To interpolating a polynomial to the data (-2,11), (-1,2),
(1,2), (2,-1), we form the Vandermonde system
1  2 4  8 a0  11 
1  1 1  1  a   2 

 1    
1 1 1 1  a2   2 

   
1
2
4
8

  a3   1


The solution is given by [1 1 1 -1].
Thus, the interpolant is p(x) = -x3 + x2 + x + 1.
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Elementary Linear Algebra
73