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T
T(u)
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u
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T . / D#
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T . C / T . / C T . /T C 90ı T
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x1
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v
u
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T(v)
u + v
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4:45 :40:25 :30:15 :15
3
5
D .x1; x2/ x1
x2 T W R2 ! R3
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2
4:45:25:15
3
5 C x2
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4:40:30:15
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T W R3 ! R3 T . / D A T . / T . /
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T . / D#
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T T 3 23 C 2
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v
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T W R2 ! R2
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A 3 # 5 TT . / D A T R3
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T T .c1 1Cc2 2/ D c1T . 1/ C c2T . 2/ 1 2
T c1 c2
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T
T W R2 ! R2
x1
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f b D 0
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f
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Rn
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T T . / D
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Au
u
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The transformation x Ax.
v
v
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D t t 0 ' t ' 1 T T .t / D t T . /T . /
TT . /
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T R2 R3
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80
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5 D
2
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3
5
T . 1/T . 2/ T . / T . /
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T . / D%
T . 1/ T . 2/& #
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x2
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A m # n j T . j / j
j Rn
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T . 1/ ! ! ! T . n/&
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T
T . / D T .x1 1 C ! ! ! C xn n/ D x1T . 1/ C ! ! ! C xnT . n/
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75 D A
A
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30
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T W R2 ! R2 R2
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A
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(1, 0)
(0, 1)
!
! x1
x2
R2
I2 1 2
x2
x1
x2
10
01
Re!ection throughthe x1-axis
01!10
!100!1
0!110
0!1!10
1001
Re!ection throughthe x2-axis
Re!ection throughthe line x2 " x1
x2 " !x1
Re!ection throughthe line x2 " !x1
!10
!10
0!1
0!1
x1
x2
Re!ection throughthe origin
x1
x2
x1
x2
10
0!1
x1
x2
01
!10
x1
x2
10
01
x2 " x1
Horizontalcontractionand expansion
Verticalcontractionand expansion
0k10
01k0
0 ! k ! 1
0 ! k ! 1 k " 1
k " 1
01
10
10
0k
01
0k
k0
k0
x1
x2
x2
x1
x2
x1
x2
x1
Horizontal shear k110
011k
k ! 0 k " 0
01
01
10
k1
k1
1k
1k
Vertical shear
k ! 0 k " 0
x2
k k 10
k
k
x2
x1
x2
x1
x2
x1x1
0100
0010
10
00
01
Projection ontothe x1-axis
Projection ontothe x2-axis
x1
x2
00
x1
x2
T W Rn ! Rm Rm Rm
Rn
T Rm T Rm
T Rn Rm Rm
T . / D T Rn Rm TRm T . / D
m
T is onto m
n
Domain Range
T is not onto m
T
n
Domain Range
m
T
T Rm
T W Rn ! Rm Rm
Rn
T Rm T . / DT
T Rm
Rn T
m
T is not one-to-one T is one-to-one
TT
mn n
0 000
RangeRangeDomain Domain
SGR2 R2 R2
R2
T
A D
2
41 "4 8 10 2 "1 30 0 0 5
3
5
T R4 R3 T
A AR3
A D T R4
R3 A D
T
T W Rn ! Rm TT . / D
P QP Q Q
P PQ
P Q
T T . / D T T . / DT
Rn
T . / D T . / D T
T . " / D T . / " T . / D " D" ¤ T . / D
T W Rn ! Rm AT
T Rn Rm A Rm
T A
PQ Q R P
R
A Rm
Rm A DT . / D T
Rn Rm
T . / D A DT A D
A
T Rn Rm
Rm A
D .x1; x2/ T . / T .x1; x2/T ..x1; x2//
T .x1; x2/ D .3x1 C x2 5x1 C 7x2 x1 C 3x2/ TT R2 R3
T . /T
A
T . / D
2
43x1 C x2
5x1 C 7x2
x1 C 3x2
3
5 D
2
4
3
5#
x1
x2
$D
2
43 15 71 3
3
5#
x1
x2
$
A
T AA
T T R3
A A 3 # 2 A R3 AA A
R3 R3
x2
The transformation T is notonto .3
e2
e1x1
x2
T
T
x3
x1
Span{a1, a2}
a1
a2
T W R2 ! R2
2 2 " :5 1 1
x2 T
1 2
A 7 # 5 T . / D AR5 R7 T T R7
TT
T W R2 ! R4 T . 1/ D .3; 1; 3; 1/ T . 2/ D ."5; 2; 0; 0/
1 D .1; 0/ 2 D .0; 1/
T W R3 ! R2 T . 1/ D .1; 3/ T . 2/ D .4; "7/T . 3/ D ."5; 4/ 1 2 3
3 # 3
T W R2 ! R2 3"=2
T W R2 ! R2 ""=4T . 1/ D .1=
p2; "1=
p2/
T W R2 ! R21
1 " 2 2 2
T W R2 ! R2
1 2 2 C 3 1
T W R2 ! R2 "3"=4
x1 T . 1/ D ."1=p
2; 1=p
2/
T W R2 ! R2 x1
x2 D x1
T W R2 ! R2
2 2 " 2 1 1
x2 D "x1
T W R2 ! R2 x2
"=2
T W R2 ! R2
x1 x2
T
T W R2 ! R2 T . 1/T . 2/
T .2; 1/
T(e1) T(e2)
x1
x2
T W R2 ! R2
A D Π1 2! 1 2 #"1
3
$
T
x1
x2
a2
a1
2
4‹ ‹ ‹‹ ‹ ‹‹ ‹ ‹
3
5
2
4x1
x2
x3
3
5 D
2
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4x1
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5
2
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5#
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x2
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2
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x1
3
5
Tx1; x2; : : :
T .x1; x2; x3; x4/ D .0; x1 C x2; x2 C x3; x3 C x4/
T .x1; x2/ D .2x2 " 3x1; x1 " 4x2; 0; x2/
T .x1; x2; x3/ D .x1 " 5x2 C 4x3; x2 " 6x3/
T .x1; x2; x3; x4/ D 2x1 C 3x3 " 4x4 .T W R4 ! R/
T W R2 ! R2
T .x1; x2/ D .x1 C x2; 4x1 C 5x2/ T . / D.3; 8/
T W R2 ! R3
T .x1; x2/ D .x1 " 2x2; "x1 C 3x2; 3x1 " 2x2/T . / D ."1; 4; 9/
T W Rn ! Rm
n # n
T W R2 ! R2
' T
T W Rn ! Rm Rm
Rn Rm
A 3 # 2 7!A
Rn Rm
Rn Rm
n # n
R2
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a 00 d
$
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T W Rn ! Rm
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T
T W R3 ! R4
T W R4 ! R3
T W Rn ! Rm A
T A
T W Rn ! Rm A
T Rn Rm A
A T W Rn ! Rm
T . / D B
m # n B AT A D B A B
T
T W Rn ! Rm Rn Rm
m n Tm n
S W Rp ! Rn T W Rn ! Rm
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; Rp c d
TT
T R5
R5
2
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3
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x2 1 " 1
10
10
10
!1
Shear transformation Re!ection through the x2-axis
x2
01
!.51
.51
x1
x2 x2
x1 x1
2 2 " :5 1 2 C :5 1 T . 2/ D 2 C :5 1
T
%T . 1/ T . 2/
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T A A
T AA R7
T
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x1 x2 x3
'x1
(!' (
x1
x1
'x1
(!' (
D x1 1
1 2 3
x2 2 x3 3
x2 x3
x1 1 C x2 2 C x3 3 D
2
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V D RI
V R #I
I1; I2 I3
RI
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1 !1 !
4 !4 !
DC
BA
1 !
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5 volts
20 volts
30 volts
I2
I1
I3
I1
RI
4I1 C 4I1 C 3I1 D .4 C 4 C 3/I1 D 11I1
AB RI 3I2
RI 11I1 " 3I2 C30
11I1 " 3I2 D 30
"3I1 C 6I2 " I3 D 5
"3I1
6I2
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"I2 C 3I3 D "25
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11I1 " 3I2 D 30
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" I2 C 3I3 D "25
I1 D I2 DI3 D "8 I3
I1
2
411"3
0
3
5
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C I2
2
4"3
6"1
3
5
"2
C I3
2
40
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3
5
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D
2
4305
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3
5
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1
I1 I1
1
2 3
R D ; R D Π1 2 3 ! D
2
4I1
I2
I3
3
5
RR D
R D
2
43000
3
5; R D
2
4050
3
5; R D
2
400
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3
5
B D
A B
I1 " I2 D 3 " 1 D I1
I2 " I3 D
0 1
2; : : : : k
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kC1 D A k k D 0; 1; 2; : : :
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k k
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k 7! kC1
AA D
B B
I1
I2
I3
I4
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2 !
4 !
3 !
3 !
4 !
1 !
1 !
20 V
30 V
10 V
20 V
10 V
1 !
5 !
1 !
2 !
4 !
I1
I2
I3
I4
4 !
1 !
2 !
2 !
20 V
40 V
10 V
30 V
3 !
1 !
4 !
2 !
3 !
10 VI4I1
I3I23 !
5 !
1 !
7 !
6 !
2 ! 20 V
40 V
30 V
4 !
4 !
I1 I4
I2 I3
I5
2 !
1 !
2 !
1 !
1 !
3 !
50 V 40 V
20 V30 V
1 ! 3 !
4 !
3 !
2 !
3 !
0
0
4
2
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k k D 1; : : : ; 20
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0º
0º
0º
20º 20º
20º 20º
1 2
4 3
(a)
Plate A
10º
10º
40º
40º
0º 0º
10º 10º
1 2
4 3
(b)
Plate B
A D
2
436 51 1352 34 740 7 1:1
3
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2
4x1
x2
x3
3
5; D
2
433453
3
5
n nn
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A D C D
A DA D
A m # n A DA Rm
ΠA !
A DA B
A D
A m # n A DRm A m
m # n AA D
Rm
n # n A nA n # n
3 # 3 A BA B
A m # n A DA D
A D
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f ; ; gR2
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A m # n m7! A
a bax D b
a b
.x; y; ´/
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R3 a b c
Three planes intersectingin a line
Three planes intersectingin a point
Three planes with nointersection
Three planes with nointersection
(a) (b)
(c) (c')
h k
x1 C 3x2 D k
4x1 C hx2 D 8
"2x1 C hx2 D 1
6x1 C kx2 D "2
4x1 " 2x2 C 7x3 D "5
8x1 " 3x2 C 10x3 D "3
A
TT
b1 b2 b3
2x1 " 4x2 " 2x3 D b1
"5x1 C x2 C x3 D b2
7x1 " 5x2 " 3x3 D b3
f 1; 2; 3g
A
TT
A
A 2 # 3 R2
A 3 # 3 R3
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f.x; y/ W y D 2xgf.x; y/ W y D x=2g
1; 2 R2
A D Π1 2! A D
a2
a1
b
x1
x2
2 # 3 AA D R3
2 # 3 AA D R3
3 # 3 AA
A
2
43
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3
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2
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3
5
a
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2
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f 1; 2; 3; 4g R5
f 1; 2; 3g 4
f 1; 2; 3gf 1; 2g Rn
f 1; 1 C 2g
1; 2; 3 R3
f 1; 2; 3g
T W Rn ! Rm
T . / D T ." / D "T W R3 ! R3
x2 D 0T .x1; x2; x3/ D .x1; "x2; x3/ T
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2
4a 0 "b0 1 0b 0 a
3
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2
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5 D
2
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