Chapter 4 Euclidean Vector Spaces

4.1 Euclidean n-Space4.2 Linear Transformations from Rn to Rm

4.3 Properties of Linear Transformations Rn to Rm

4.1 Euclidean n-Space

DefinitionVectors in n-Space

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 n-space and is denoted by Rn

Definition Two vectors u=(u1 ,u2 ,…,un) and v=(v1 ,v2 ,…, vn) in Rn are

called equal if

The sum u+v is defined by

and if k is any scalar, the scalar multiple ku is defined by

nn v u,..., v, uvu 2211

),...,, 2211 nn vuvuv(u vu

),...,,( 21 nkukukuk u

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)

Theorem 4.1.1Properties 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:

(a) u+v = v+u (b) u+(v+w) = (u+v)+w(c) u+0 = 0+u = u (d) u+(-u) = 0; that is u-u = 0(e) k(lu) = (kl)u (f) k(u+v) = ku+kv(g) (k+l)u = ku+lu (h) 1u = u

Definition Euclidean Inner Product

If u=(u1 ,u2 ,…,un), v=(v1 ,v2 ,…, vn) are vectors in Rn , then the Euclidean inner product u٠v is defined by

nnvuvuvu ...2211vu

Example 1Inner Product of Vectors in R4

The Euclidean inner product of the vectors

u=(-1,3,5,7) and v=(5,-4,7,0)in R4 is

u٠v=(-1)(5)+(3)(-4)+(5)(7)+(7)(0)=18

Theorem 4.1.2Properties of Euclidean Inner Product

If u, v and w are vectors in Rn and k is any scalar, then(a) u٠v = v٠u (b) (u+v)٠w = u٠w+ v٠w

(c) (k u)٠v = k(u٠v)

(d) Further, if and only if v=0

0vv 0vv

Example 2Length and Distance in R4

(3u+2v)٠(4u+v) = (3u)٠(4u+v)+(2v)٠(4u+v)

= (3u)٠(4u)+(3u)٠v

+(2v)٠(4u)+(2v)٠v

=12(u٠u)+11(u٠v)+2(v٠v)

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

Similarly, the Euclidean distance between the points u=(u1 ,u2 ,…,un) and v=(v1 , v2 ,…,vn) in Rn is defined by

222

21

21

...( nuuu) uuu

2222

211 )(...)()(),( nn vuvuvud vuvu

Example 3Finding Norm and Distance If u=(1,3,-2,7) and v=(0,7,2,2), then in the

Euclidean space R4

2222

2222

)27()22()73()01(),(

7363)7()2()3()1(

vu

u

d

and

Theorem 4.1.3Cauchy-Schwarz Inequality in Rn

If u=(u1 ,u2 ,…,un) and v=(v1 , v2 ,…,vn) are vectors in Rn, then

vuvu

Theorem 4.1.4Properties of Length in Rn

If u and v are vectors in Rn and k is any scalar, then

)inequality (Triangle (d)

(c)

ifonly and if 0 (b)

0 (a)

vuvu

uu

0uu

u

kk

Theorem 4.1.5Properties of Distance in Rn

If u, v, and w are vectors in Rn and k is any scalar, then:

)inequality (Triangle ),(),(),( (d)

),(),( (c)

ifonly and if 0),( (b)

0),( (a)

vwwuvu

uvvu

vuvu

vu

ddd

dd

d

d

Theorem 4.1.6 If u, v, and w are vectors in Rn with

the Euclidean inner product, then

22

4

1

4

1vuvuvu

Definition Orthogonality

Two vectors u and v in Rn are called orthogonal if u٠v=0

Example 4Orthogonal Vector in R4

0)1)(4()0)(1()2)(3()1)(2(

since ,orthogonal are

)1 ,0 ,2 ,1( and )4 ,1 ,3 ,2(

vectors the spaceEuclidean In the 4

vu

vu

R

Theorem 4,1,7Pythagorean Theorem in Rn

thenproduct,inner Euclidean which theR

in vectorsorthogonal areand If

222

n

vuvu

v u

Alternative Notations for Vectors in Rn (1/2)

nnnnnn

n

n

n

ku

ku

ku

u

u

u

k k

uu

uu

uu

v

v

v

u

u

u

uuu

u

u

u

uuu

,

... or

matrixcolumn aor matrix row a asnotation matrix in R

in ),...,,( vector a write tousefuloften isIt

2

1

2

1

22

11

2

1

2

1

212

1

n

21

uvu

uu

u

Alternative Notations for Vectors in Rn (2/2)

)(

) ..., , ,(), ... , ,(

operation vector theas results same theproduce

...

... ...

or

2211

2121

2121

1211

2121

nn

nn

nn

nn

nn

v, ..., uv, uvu

vvvuuu

... ku kukuuuukk

v ... uv uvu

vvvuuu

vu

u

vu

A Matrix Formula for the Dot Product(1/2)

)(

v u ...v uvu

u

u

u

... v vv

v

v

v

u

u

u

nn

n

n

nn

7

productinner Euclidean for the formula

following thehave enotation wmatrix column in sfor vector Thus,

and

vectorsfor thenotation matrix column use weIf

22112

1

21

2

1

2

1

uvvu

vuvu

uv

vu

T

T

A Matrix Formula for the Dot Product(2/2)

)(AA

)(AA

AAAAA

A

nnA

T

T

TTTTTT

TTTT

9

8

(((

A(A)()A(A

that

transpose theof properties and (7) formula

form followsit then matrix, a is If

T

vuvu

vuvu

vuu)v)uvuv)vu

vuuv)uvuvvu

Example 5Verifying That vuvu TAA

11)1(4)4(2)7)(1(

11)5(5)0(10)2(7

1

4

7

5

0

2

1 1 3

0 4 2

1 2 1

5

10

7

4

2

1

101

142

321

Then

5

0

2

,

4

2

1

,

101

142

321

thatSuppose

vu

vu

v

u

vu

T

T

A

A

A

A

A

A Dot Product View of Matrix Multiplication (1/2)

rj

j

j

irii

rjirjiji

ijij

b

b

b

Bj

... a aa

Ai

bababa

ABij

nrbBrmaA

2

1

21

2211

oftor column vecth theand

of vector rowth theofproduct dot theisWhich

...

is ofentry th then the

matrix, an is andmatrix an is If

A Dot Product View of Matrix Multiplication (2/2)

brrr

xr

xr

xr

b x

crcrcr

crcrcr

crcrcr

ccc

rrr

of entries theare ,...,, and , of vectorsrow theare ..., where

(11)

as formproduct dot in expressed becan systemlinear A

(10)

as expressed becan product matrix then the,...,

are of torscolumn vec theand ..., are of vectorsrow theif Thus,

2121

2

1

2

1

21

22212

12111

21

21

mm

mm

nmmm

n

n

n

m

bbbA,,

b

b

b

A

AB

AB,,

B,,A

Example 6 A Linear System Written in Dot Product Form

085

5472

143

321

321

321

xx x

xxx

xxx

(11) from expressed systemlinear a of examplean is following The

0

5

1

),,()8,5,1(

),,()4,7,2(

),,()1,4,3(

321

321

321

xxx

xxx

xxx

System Dot Product Form

4.2 Linear Transformations From Rn to Rm

Functions from Rn to RFormula Example Classification Description

Real-valued function of a real variable

Function from 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

)(xf 2)( xxf

),( yxf22),( yxyxf

),,( zyxf 22

2

),,(

zy

xzyxf

),...,,( 21 nxxxf22

221

21

...

),...,,(

n

n

xxx

xxxf

Functions from Rn to Rm

(1/2) 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 , and we say that the function f maps Rn into Rm. We denote this by writing f :

In the case where m=n the transformation f : is called an operator on Rn

mn RR

mn RR

Functions from Rn to Rm

(2/2)

Suppose that 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 assign a unique point (w1,w2,…,wm) in Rm to each point (x1,x2,…,xn) in Rn and thus define a transformation from Rn to Rm. If we denote this transformation by T: then

T (x1,x2,…,xn)= (w1,w2,…,wm)

mn RR

Example 1 32 R toR fromtion TransformaA

)3,6,1()2,1( example,for Thus,

),3,()

.ation transforma define

3

equations The

22

21212121

32

22

213

212

211

T

xxxxxx,xT(x

RT:R

xxw

xxw

xxw

mn RR to from tionsTransformaLinear

nmnmmm

nn

nn

xaxaxaw

xaxaxaw

xaxaxaw

...

...

...

2211

22221212

12121111

xw A

aaa

aaa

aaa

w

w

mmm

by briefly moreor

w

notationmatrix in or

321

232221

131211

m

2

1

The transformation define by those equations is called a linear transformation ( or a linear operator if m=n ). Thus, a linear transformation is defined by equations of the form

mn RRT

mn RRT

The matrix A=[aij] is called the standard matrix for the linear transformation T, and T is called multiplication by A

Example 2A Linear Transformation from R4 to R3

8

3

1

2

0

3

1

0 4 1 5

1 2 1 4

5 1 3 2

8w,3w,1 wyields (5)in ngsubstituti then ),2,0,3,1(),,,(x

if example,For tion.multiplicamatrix by (6) fromor (5) equations

fromdorectly computed becan ),,,(xpoint a of image The

(6)

0 4 1 5

1 2 1 4

5 1 3 2

as formmatrix in expressed becan

45

(5) 24

532

equations by the defined n nsformatiolinear tra The

3

2

1

3214321

4321

4

3

2

1

3

2

1

3213

43212

43211

34

w

w

w

xxx

xxx

x

x

x

x

w

w

w

xxxw

xxxxw

xxxxw

RT:R

0 4 1 5

1 2 1 4

5 1 3 2

is Tfor matrix standard the

A

Some Notational Matters We denote the linear transformation by

Thus,

The vector is expressed as a column matrix.

We will denote the standard matrix for T by the symbol [T].

Occasionally, the two notations for standard matrix will be mixed, in which case we have the relationship

mn RRT mnA RRT

xx ATA )(

xx ][)( TT

ATA ][

x

Example 3Zero Transformation from Rn to Rm

0.by denoted isation transformzero theSometimes

. to fromation transformzero the call We.in vector

zero theinto in or every vect maps zeroby tion multiplica so

0)(

in or every vectfor

then,in vector zero theis andmatrix zero theis 0 If

0

0

mnm

n

n

n

RRTR

R

T

R

Rnm

0xx

x

0

Example 4Identity Operator on Rn

nI

n

n

RT

RI

IT

Rnn

on operator identity the call We

itself. into in or every vect maps by tion multiplica so

)(

in or every vectfor then identity, theis I If

I xxx

x

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.

Tables 2 and 3 list some of the common reflection operators

Table 2

Table 3

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. It can be shown that operators are linear.

Some of the basic projection operators on R2 and R3 are listed in Tables 4 and 5.

Table 4

Table 5

Rotation Operators (1/2) An operator that rotate

each vector in R2 through a fixed angle is called a rotation operator on R2. Table 6 gives formulas for the rotation operator on R2.

Consider the rotation operator that rotates each vector counterclockwise through a fixed angle . To find equations relating and ,let be the positive -axis to ,and let r be the common length of and (figure 4.2.4)

xx

x

w

(x)w T x

Rotation Operators (2/2)

θθ

θθT

T

θyθxw

θyθxw

θrθr w

θrθrw

) (θr), w(θrw

r, yrx

cos sin

sin cos][

for matrix standard The

cossin

(16) sincos yields (14) ngsubstituti and

sincoscossin

sinsincoscos

yields, (15)on identitiestrix trigonomeUsing

(15) sincos

(14) sincos

try trigonomebasic fromThen

2

1

1

1

21

Table 6

Example 5Rotation

2

31

2

13

is 1

1

vector theof image theexample,For

23

21

21

23

23 2

1

21 2

3

6 cos 6sin

6sin 6cos is

vector a of image then the,)30(6 of anglean through rotated is in r each vecto If 2

wx

w

x

w

yx

yx

y

x

y

x

y

x

R

A Rotation of Vectors in R3(1/3) 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 (figure 4.2.5a). 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.

For example, in figure 4.2.5a , angles are positive if they are generated by counterclockwise rotations and negative if they are generated by clockwise.

The most common way of describing a general axis of rotation is to specify a nonzero vector u that runs along the axis of rotation and has its initial point at the origin. The counterclockwise direction for a rotation about its axis can be determined by a “right-hand rule” (Figure 4.2.5 b)

A Rotation of Vectors in R3(2/3) A rotation operator on R3 is a linear operator

that rotates each vector in R3 about some rotation axis through a fixed angle .

In table 7 we have described the rotation operators on R3 whose axes of rotation are positive coordinate axes.

Table 7

A Rotation of Vectors in R3(3/3) We note that the standard matrix for a

counterclockwise rotation through an angle about an axis in R3, which is determined by an arbitrary unit vector that has its initial point at the origin, is

cos)cos1( sin)cos1(c sin)cos1(

sin)cos1(c cos)cos1( sin)cos1(

sin)cos1( sin)cos1( cos)cos1(

2

2

2

cabbac

abbcab

baccaba

),,( cbau

Dilation and Contraction Operators If is a nonnegative scalar, the operator

on R2 or R3 is called a contraction with factor if

and a dilation with factor if .

Table 8 and 9 list the dilation and contraction operators on R2 and R3

10 k 1k

kk

k

k

Table 8

Table 9

Compositions of Linear Transformations

(21) ]][[][

:(20) formula of form ealternativan is There

(20)

(19)())(())((

sincelinear is n compositio The

(18) ))(())((

Thus )." circle " (read by denoted

is and with ofn compositio thecalled isation transformThis .

tofromation transforma produces by followed ofn applicatio

theThus, .in vector a iswhich ,))(( computecan onethen

and ,in vector a is which ),( computefirst can one in each

for then ns,nsformatiolinear tra are and If

1212 TTTT

TTT

BAABTTTT

TT

TTTT

TTTT

TTR

RTT

RTT

RTR

RRTRR T

BAAB

ABAB

AB

ABAB

ABAB

ABn

mBA

mAB

kA

n

mkB

knA

)x (x)xx

xx

x

xx

Example 6Composition of Two Rotations(1/2)

Let and be linear operators that rotate vectors through the angle and ,respective. Thus the operation

first rotates through the angle , then rotates through the angle . It follows that the net effect of

is to rotate each vector in R2 through the angle (figure 4.2.7)

221 : RRT 22

2 : RRT

1

2

))((())(( 1212 xx TTTT

1

2

x)(1 xT

21 12 TT

Example 6Composition of Two Rotations(2/2)

][

)cos( )sin(

)sin( )cos(

coscossinsin sincoscossin

)cossinsin(cos sinsincoscos

cos sin

sin cos

cos sin

sin cos]][[

(21).satisfy should matrices These

)cos( )sin(

)sin( )cos(][

cos sin

sin cos][ ,

cos sin

sin cos][

are operatorslinear for these matrices standard theThus,

12

2121

2121

12121212

12121212

11

11

22

2212

2121

212112

22

222

11

111

TT

TT

TT

TT

Example 7Composition Is Not Commutative(1/2)

Let be the reflection operator about the line ,and let be the orthogonal projection on the -axis. Figure 4.2.8 illustrates graphically that and have different effect on a vector . This same conclusion can be reached by showing that the standard matrices for and do not commute:

221 : RRT

222 : RRT

12 TT 21 TT

xy

y

x

1T 2T

Example 7Composition Is Not Commutative(2/2)

1221

1212

2121

so

0 1

0 0

0 1

1 0

0 0

1 0]][[

0 0

1 0

0 0

1 0

0 1

1 0]][[

TTTT

TTTT

TTTT

Example 8Composition of Two Reflections(1/2)

Let be the reflection about the -axis, and let be the reflection about the -axis. In this case and are the same; both map each vector into negative

(Figure 4.2.9)

221 : RRT 22

2 : RRT y

x21 TT 12 TT

),( yxx

),( yx x

),(),(),)((

),(),(),)((

212

121

yxyxTyxTT

yxyxTyxTT

Example 8Composition of Two Reflections(2/2)

The equality and can also be deduced by showing that the standard matrices for and commute

The operator on R2 or R3 is called the reflection about the origin. As the computations above show, the standard matrix for this operator on R2 is

21 TT 12 TT

1T 2T

1 0

0 1

1 0

0 1

1 0

0 1]][[][

1 0

0 1

1 0

0 1

1 0

0 1]][[][

1212

2121

TTTT

TTTT

xx )(T

1 0

0 1T

Compositions of Three or More Linear Transformations Compositions can be defined for three or more linear

transformations. For example, consider the linear transformations

We define the composition by

It can be shown that this composition is a linear transformation and that the standard matrix for is related to the standard matrices for , , and by

which is a generalization of (21) . If the standard matrices for , , and are denoted by A, B, and C, respectively, then we also have the following generalization of (20):

mllkkn RRTRRTRRT : ,: ,: 321

mn RRTTT :)( 123

)))(((:)( 123123 xx TTTTTT

]][][[][ 123123 TTTTTT

CBAABC TTTT

123 TTT

1T

1T

2T

3T

3T

2T

Example 9Composition of Three Transformations(1/2)

Find the standard matrix for the linear operator that first rotates a vector counterclockwise about the -axis through an angle , then reflects the resulting vector about the -plane, and then projects that vector orthogonally onto the -plane.

Solution:The linear transformation T can be expressed as the composition , where T1 is the rotation about the -axis, T2 is the rotation about the -plane, T3 is the rotation about the -plane. From Tables 3,5, and 7 the standard matrices for these linear transformations are

33: RRT

z

yz

xy

123: TTTT yz

xy

Example 9Composition of Three Transformations(2/2)

0 0 0

0 cos sin

0 sin cos

1 0 0

0 cos sin

0 sin cos

1 0 0

0 1 0

0 0 1

0 0 0

0 1 0

0 0 1

][

is that ,[ is Tfor matrix standard the(22) from Thus,

0 0 0

0 1 0

0 0 1

][ ,

1 0 0

0 1 0

0 0 1

][ ,

1 0 0

0 cos sin

0 sin cos

][

123

321

T

]][T][T[TT]

TTT

4.3 Properties of Linear Transformations from Rn to Rm

DefinitionOne-to-One Linear transformations

A linear transformation T=Rn →Rm is said to be one-to-one if T maps distinct vectors (points) in Rn into distinct vectors (points) in Rm

Example 1One-to-One Linear Transformations

In the terminology of the preceding definition, the rotation operator of Figure 4.3.1 is one-to-one, but the orthogonal projection operator of Figure 4.3.2 is not

Theorem 4.3.1Equivalent Statements

If A is an nxn matrix and TA: Rn→Rn is multiplication by A, then the following statements are equivalent.

(a) A is invertible(b) The range of TA is Rn

(c) TA is one-to-one

Example 2Applying Theorem 4.3.1 In Example 1 we observed that the rotation operator T: R2→R2

illustrated in Figure 4.3.1 is one-to-one. It follows from Theorem 4.3.1 that the range of T must be all of R2 and that the standard matrix for T must be invertible. To show that the range of T is all of R2, we must show that every vector w in R2 is the image of some vector x under T. But this is clearly so, since the vector x obtained by rotating w through the angle - maps into w when rotated through the angle . Moreover, from Table 6 of Section 4.2, the standard matrix for T is

Which is invertible, since

cos sin

sin cos][

T

01sincoscos sin

sin cosdet 22

Example 3Applying Theorem 4.3.1 In Example 1 we observed that the projection operator T:

R3→R3 illustrated in Figure 4.3.2 is not one-to-one. It follows from Theorem 4.3.1 that the range of T is not all of R3 and the standard matrix for T is not invertible. To show that the range of T is not all of R3, we must find a vector w in R3 that is not the image of any vector x under T. But any vector w outside of the xy-plane has this property, since all images under T lie in the xy-plane. Moreover, from Table 5 of Section 4.2, the standard matrix for T is

which is not invertible, since det[T]=0

0 0 0

0 1 0

0 0 1

][T

Inverse of a One-to-One Linear Operator(1/2) If TA=Rn →Rn is a one-to-one linear operator, then from

Theorem 4.3.1 the matrix A is invertible. Thus, is itself a linear operator; it is called the inverse of TA. The linear operators TA and cancel the effect of one another in the sense that for all x in Rn

or equivalently,

If w is the image of x under TA, then maps w back into x, since

nn

ARRT 1

1AT

1AT

xxxx

xxxx

IAATT

IAATT

AA

AA

1

1

))((

))((

1

1

IAAAA

IAAAA

TTTT

TTTT

11

11

xxw ))(()( 11 AAATTT

Inverse of a One-to-One Linear Operator(2/2) 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. since the standard matrix for T-1 is the inverse of the standard matrix for T, we have

[T-1]=[T]-1

Example 4Standard Matrix for T-1

Let T: R2→R2 be the operator that rotates each vector in R2 through the angle ;so from Table 6 of Section 4.2

It is evident geometrically that to undo the effect of T one must rotate each vector in R2 through the angle .But this is exactly what the operator T-1 does, since the standard matrix T-1 is

,which is identical to (2) except that is replaced by

cos sin

sin cos][T

)cos( )sin(

)sin( )cos(

cos sin

sin cos ][][ 11

TT

Example 5Finding T-1 (1/2) 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:The matrix form of these equations is

so the standard matrix for T is

2

1

2

1

4 3

1 2

x

x

w

w

4 3

1 2][T

Example 5Finding T-1 (2/2) This matrix is invertible (so T is one-to-one) and the

standard matrix for T-1 is

Thus,

from which we conclude that

5

2

5

35

1

5

4

][][ 11 TT

21

21

2

1

2

11

5

2

5

35

1

5

4

5

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Theorem 4.3.2Properties of Linear Transformations

A 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

(a) T(u+v) = T(u) + T(v)(b) T(cu) = cT(u)

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

[T]=[T(e1)|T(e2)|…|T(en)] (6)

Example 6Standard Matrix for a Projection Operator(1/3)

Let l be the line in the xy-plane that passes through the origin and makes an angle with the positive x-axis, where . As illustrated in Figure 4.3.5a, let T: R2→R2 be a linear operator that maps each vector into orthogonal projection on l.(a) Find the standard matrix for T(b) Find the orthogonal projection of the vector x=(1,5) onto the line through the origin that makes an angle of with the positive x-axis

Solution (a): From (6)[T]=[T(e1)|T(e2)]

where e1 and e2 are the standard basis vectors for R2. We consider the case where ; the case where is similar.

0

6

20 2

Figure 4.3.5

Example 6Standard Matrix for a Projection Operator(2/3)

Referring to Figure 4,3,5b, we have , so

and referring to Figure 4.3.5c, we have so

thus, the standard matrix for T is

cos)( 1 eT

sin)( 2 eT

cossin

cos

sin)(

cos)()(

2

1

1

1e

ee

T

TT

2

2

2

2sin

cossin

sin)(

cos)()(

e

ee

T

TT

2

2

sin cossin

cossin cos T

Example 6Standard Matrix for a Projection Operator(3/3)

Solution (b):Since , it follows from part (a) that the standard matrix for this projection operator is

thus,

or in horizontal notation

236cos and 216sin

41 43

43 43][T

4

53

4

353

5

1

41 43

43 43

5

1T

4

53,

4

3535,1T

Definition 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

Those nonzero vectors x that satisfy this equation are called the eigenvectors of T corresponding to

(7) )( xx T

Example 7Eigenvalues of a Linear Operator(1/3)

Let T: R2→R2 be the linear operator that rotates each vector through an angle . It is evident geometrically that unless is a multiple of , then T does not map any nonzero vector x onto the same line as x; consequently, T has no real eigenvalues. But 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. Let us verify these geometric observations algebraically. The standard matrix for T is

As discussed in Section 2.3, the eigenvalues of this matrix are the solutions of the character equation

cos sin

sin cos

A

cos sin

sin cos)det(

AI

Example 7Eigenvalues of a Linear Operator(2/3)

That is,

But if is not a multiple of , then , so this equation has no real solution for and consequently A has no real eigenvectors. If is a multiple of , then and either or , depending on the particular multiple of .In the case where and ,the characteristic equation (8) becomes , so

is the only eigenvalue of A. In the case the matrix A is

Thus, for all x in R2

0sin)cos( 22

0sin 2

0sin 0sin

1cos

1cos 1cos

0)1( 2 1

1 0

0 1A

xxxx IAT )(

Example 7Eigenvalues of a Linear Operator(3/3)

So T maps every vector to itself, and hence to the same line. In the case where and , the characteristic equation (8) becomes , so that is the only eigenvalue of A. In this case the matrix A is

Thus, for all x in R2 ,

so T maps every vector to its negative, and hence to the same line as x.

0sin 1cos

IA

1 0

0 1

0)1( 2 1

xxxx IAT )(

Example 8 Eigenvalues of a Linear Operator(1/3) Let T: R3→R3 be the orthogonal projection on xy-plane. Vectors

in the xy-plane are mapped into themselves under T, so each nonzero vector in the xy-plane is an eigenvector corresponding to the eigenvalue .Every vector x along the z-axis is mapped into 0 under T, which is on the same line as x, so every nonzero vector on the z-axis is an eigenvector corresponding to theei genvalue . Vectors not in the xy-plane or along the z-axis are mapped into scalar multiples of themselves, so there are no other eigenvectors or eigenvalues.

To verify these geometric observations algebraically, recall from Table 5 of Section 4.3 that the standard matrix for T is

1

0 0 0

0 1 0

0 0 1

A

0

Example 8 Eigenvalues of a Linear Operator(2/3)

The characteristic equation of A is

which has the solutions and anticipated above.As discussed in Section 2.3, the eigenvectors of the matrix A corresponding to an eigenvalue are the nonzero solutions of

If , this system is

0)1(or 0

0 0

0 1 0

0 0 1

)det( 2

AI

1

(9)

0

0

0

0 0

0 1 0

0 0 1

3

2

1

x

x

x

0

0

0

0 0 0

0 1 0

0 0 1

3

2

1

x

x

x

0

0

Example 8 Eigenvalues of a Linear Operator(3/3)

which has the solutions x1=0,x2=0,x3=t (verify), or in matrix form

As anticipated, these are the vectors along the z-axis. If , then system (9) is

which has the solutiona x1=s , x2=t , x3=0 , or in matrix form,

As anticipates, these are vectors in xy-plane

tx

x

x

0

0

3

2

1

03

2

1

t

s

x

x

x

1

0

0

0

1 0 0

0 0 0

0 0 0

3

2

1

x

x

x

Theorem 4.3.4Equivalent Statements If A is an nxn matrix, and if TA: Rn→Rn is multiplication by

A , then the following are equivalent,

(a) A is invertible(b) Ax=0 has only the trivial solution(c) The reduced row-echelon form of A is In

(d) A is expressible as a product of elementary matrices(e) Ax=b is consistent for every nx1 matrix b(f) Ax=b has exactly one solution for every nx1 matrix b(g) (h) The range of TA is Rn

(i) TA is one-to-one

0)det( A