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2u
u
The set of all multiples of uTypical multiples of u
23
– –
R3
R3 3 # 1
2a
a
x2x1
x3
D
2
4234
3
5 2
Rn
n Rn
n n n # 1
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2
6664
u1
u2
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3
7775
Rn
Rn R2
Rn
; ; Rn c d
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C D C D c.d / D .cd/
C ." / D " C D 1 D" ."1/
C ."1/ "" "
x1
x2
v
u
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u – v
1; 2; : : : ; p Rn c1; c2; : : : ; cp
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1; : : : ; p c1; : : : ; cp
1 2p3 1 C 2; 1
2 1 .D 12 1 C 0 2/; .D 0 1 C 0 2/
1 D#
"11
$
2 D#
21
$
1 2 1 2
3v2
2v22v1
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v1 – v2 –2v1 + v2
3v1
v1
–v1–v2
wu
v2
v1 + v2 3v2
32–
0
1 2
3 1 "2 2
D 3 1 " 2 2
1 3 1 "2 2
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v1
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0
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1 D
2
41
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3
52 D
2
4256
3
5 D
2
474
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3
5
1 2
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x1 1 C x2 2 D
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2
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3
5
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C x2
2
4256
3
5
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D
2
474
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3
5
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2
4x1
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3
5 C
2
42x2
5x2
6x2
3
5 D
2
474
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3
5
2
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3
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2
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3
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3
5 %
2
41 2 70 9 180 16 32
3
5 %
2
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3
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3
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3
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2
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f 1; : : : ; pg
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v
f g
Span{u, v}
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h! 5v3
v1
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lie on a line that
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Span {v1, v2, v3}
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2
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3
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3
5
h " 5 f 1; 2; 3g h D 5
f 1; 2; 3g c1 c2 c3
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Span{a1, a2, a3}
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u
x2
T W R2 ! R2
T . / D#
0 "11 0
$#x1
x2
$D
#"x2
x1
$
T D#
41
$D
#23
$C D
#64
$
T . / D#
0 "11 0
$#41
$D
#"1
4
$; T . / D
#0 "11 0
$#23
$D
#"3
2
$;
T . C / D#
0 "11 0
$#64
$D
#"4
6
$
T . C / T . / C T . /T C 90ı T
T . / T . /
T
x1
x2
v
u
T(u + v)
T(u)
T(v)
u + v
U D Π!
U D
2
4:45 :40:25 :30:15 :15
3
5
D .x1; x2/ x1
x2 T W R2 ! R3
T . / D U D x1
2
4:45:25:15
3
5 C x2
2
4:40:30:15
3
5 D
2
4
3
5
T
4T . / 4T . /
C T . /T . /
T W R5 ! R2 T . / D A A R5
A
A D#
1 00 "1
$7! A
t0 ' t ' 1 T
T . /
A D#
2 00 2
$T W R2 ! R2 T . / D A
T D#
1"3
$D
#ab
$
A D
2
4:5 0 00 :5 00 0 :5
3
5 D
2
410
"4
3
5 D
2
4abc
3
5
T W R3 ! R3 T . / D A T . / T . /
T T . / D AT
A D
2
41 0 "2
"2 1 63 "2 "5
3
5 D
2
4"1
7"3
3
5
A D
2
41 "3 20 1 "43 "5 "9
3
5 D
2
46
"7"9
3
5
A D#
1 "5 "7"3 7 5
$D
#"2"2
$
A D
2
664
1 "2 13 "4 50 1 1
"3 5 "4
3
775 D
2
664
193
"6
3
775
A 6 # 5 a bT W Ra ! Rb T . / D A
AR4 R5 T . / D A
R4
7!A A
A D
2
41 "4 7 "50 1 "4 32 "6 6 "4
3
5
A D
2
664
1 3 9 21 0 3 "40 1 2 3
"2 3 0 5
3
775
D
2
4"1
10
3
5 A
7!A
D
2
664
"13
"14
3
775 A
7!A
D#
52
$D
#"2
4
$
TT
R2
T . / D#
"1 00 "1
$ #x1
x2
$
T . / D#
:5 00 :5
$ #x1
x2
$
T . / D#
0 00 1
$ #x1
x2
$
T . / D#
0 11 0
$ #x1
x2
$
T W R2 ! R2
D#
52
$ #21
$D
#13
$ #"1
3
$
T T 3 23 C 2
T . / T . /T W R2 ! R2
T . /
uw
v
T(v)
T(u)
x2 x2
x1x1
1 D#
10
$2 D
#01
$1 D
#25
$2 D
#"1
6
$
T W R2 ! R21
1 2 2
#5
"3
$
#x1
x2
$
D#
x1
x2
$1 D
#"2
5
$2 D
#7
"3
$
T W R2 ! R2
x1 1 C x2 2 A T . / A
A 3 # 5 TT . / D A T R3
A m # n7!A Rm
T T .c1 1Cc2 2/ D c1T . 1/ C c2T . 2/ 1 2
T c1 c2
7!AA
T W Rn ! Rm
Rm
T
T W R2 ! R2
x1
1; : : : ; p Rn T W Rn ! Rn
T . i / D i D 1; : : : ; pT
Rn T . / D¤ Rn
D C tT W Rn ! Rn
R3 P
P D s C t s; t RT W R3 ! R3 P
R3
T . / T . /P
Rn
D .1 " t/ C t
.1 " t/ C t 0 ' t ' 1T
(t = 1) q (1 – t)p + tqx
(t = 0) p
Rn P
a C b 0 ' a ' 1 0 ' b ' 1 T W Rn ! Rm
P TT . / T . /
f W R ! R f .x/ D mx C b
f b D 0
b ¤ 0
f
T W Rn ! Rm
T .x/ D A C A m # n Rm
T ¤
T W Rn ! Rm
f 1; 2; 3g Rn
fT . 1/; T . 2/; T . 3/g
D .x1; x2/ T . / T .x1; x2/
T T .x1; x2/ D.4x1 " 2x2; 3jx2j/
T T .x1; x2/ D.2x1 " 3x2; x1 C 4; 5x2/
T W Rn ! Rm
TT . / D
Rn
T . / T . /c1T . / C c2T . / D c1 c2
T W R3 ! R3
D .x1; x2; x3/ x3 D 0T . / D .x1; x2; "x3/ T
T W R3 ! R3
D .x1; x2; x3/ x2 D 0 T . / D.x1; 0; x3/ T
T T . / D
2
664
4 "2 5 "5"9 7 "8 0"6 4 5 3
5 "3 8 "4
3
775
2
664
"9 "4 "9 45 "8 "7 67 11 16 "99 "7 "4 5
3
775
D
2
664
7597
3
775 A
7!A
D
2
664
"7"713"5
3
775 A
7!A
SG
A A A
Au
u
x2
x1
A
The transformation x Ax.
v
v
Ax
xT R2
.4; 1/ .4; "1/ 7!Ax x1
D t t 0 ' t ' 1 T T .t / D t T . /T . /
TT . /
Rn Rm 7!AT A
A Tn # n In
I2 D#
1 00 1
$1 D
#10
$2 D
#01
$
T R2 R3
T . 1/ D
2
45
"72
3
5 T . 2/ D
2
4"3
80
3
5
R2
x1
x2
e2 = 01
!"
!"
e1 = 10
!"
!"
D#
x1
x2
$D x1
#10
$C x2
#01
$D x1 1 C x2 2
T
T . / D x1T . 1/ C x2T . 2/
D x1
2
45
"72
3
5 C x2
2
4"3
80
3
5 D
2
45x1 " 3x2
"7x1 C 8x2
2x1 C 0
3
5
T . 1/T . 2/ T . / T . /
A
T . / D%
T . 1/ T . 2/& #
x1
x2
$D A
T W Rn ! Rm
AT . / D A Rn
A m # n j T . j / j
j Rn
A D%
T . 1/ ! ! ! T . n/&
D In D Π1 ! ! ! n ! D x1 1 C ! ! ! C xn n
T
T . / D T .x1 1 C ! ! ! C xn n/ D x1T . 1/ C ! ! ! C xnT . n/
D%
T . 1/ ! ! ! T . n/&
2
64x1
:::xn
3
75 D A
A
AT
Rn Rm
A T . / D 3R2
T . 1/ D 3 1 D#
30
$T . 2/ D 3 2 D
#03
$
# #
A D#
3 00 3
$
T W R2 ! R2 R2
'
A
#10
$ #''
$ #01
$ #" '
'
$
A D#
' " '' '
$
' D "=2
(– sin !, cos !)
(cos !, sin !)
(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
43x1 " 2x3
4x1
x1 " x2 C x3
3
5
2
4‹ ‹‹ ‹‹ ‹
3
5#
x1
x2
$D
2
4x1 " x2
"2x1 C x2
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
R2#
a 00 d
$
a d ˙1
T W Rn ! Rm
Rn Rm
A 3 # 2 7! AR2 R3
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
7! T .S. //Rp Rm T .S.c C d //
; Rp c d
TT
T R5
R5
2
664
"5 10 "5 48 3 "4 74 "9 5 "3
"3 "2 5 4
3
775
2
664
7 5 4 "910 6 16 "412 8 12 7"8 "6 "2 5
3
775
2
66664
4 "7 3 7 56 "8 5 12 "8
"7 10 "8 "9 143 "5 4 2 "6
"5 6 "6 "7 3
3
77775
2
66664
9 13 5 6 "114 15 "7 "6 4"8 "9 12 "5 "9"5 "6 "8 9 813 14 15 2 11
3
77775
1 2 1WEB" 1 T . 1/ D " 1 2 2 " :5 1
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/
&D
%" 1 2 C :5 1
&D
#"1 :5
0 1
$
T A A
T AA R7
T
WEB
: : : :
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
436 51 13 3352 34 74 450 7 1 3
3
5 % ! ! ! %
2
41 0 0 :2770 1 0 :3920 0 1 :233
3
5
x1 x2 x3
":233
WEB
V D RI
V R #I
I1; I2 I3
RI
1 !1 !
1 !1 !
4 !4 !
DC
BA
1 !
3 !
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
"I3 D "1 ! I3
"I2 C 3I3 D "25
"5
11I1 " 3I2 D 30
"3I1 C 6I2 " I3 D 5
" I2 C 3I3 D "25
I1 D I2 DI3 D "8 I3
I1
2
411"3
0
3
5
"1
C I2
2
4"3
6"1
3
5
"2
C I3
2
40
"13
3
5
"3
D
2
4305
"25
3
5
"
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
"25
3
5
B D
A B
I1 " I2 D 3 " 1 D I1
I2 " I3 D
0 1
2; : : : : k
kA 1 D A 0 2 D A 1
kC1 D A k k D 0; 1; 2; : : :
1 2 0
k
k k
r0 s0 0
0 D#
r0
s0
$
1 D#
r1
s1
$; 2 D
#r2
s2
$; 3 D
#r3
s3
$; : : :
.03
.05
.95 .97
City Suburbs
r0
#:95r0
:05r0
$D r0
#:95:05
$
s0
s0
#:03:97
$
#r1
s1
$D r0
#:95:05
$C s0
#:03:97
$D
#:95 :03:05 :97
$#r0
s0
$
1 D M 0
M
#:95:05
:03:97
$
2 D M 1
kC1 D M k k D 0; 1; 2; : : :
f 0; 1; 2; : : :g
0 D#
600;000400;000
$
1 D#
:95 :03:05 :97
$#600;000400;000
$D
#582;000418;000
$
2 D M 1 D#
:95 :03:05 :97
$#582;000418;000
$D
#565;440434;560
$
k 7! kC1
AA D
B B
I1
I2
I3
I4
1 !
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
4:97:00:03
:05:90:05
:10:05:85
3
5
M 0
k k D 1; : : : ; 20
T1 T2 T3 T4
Tk
.20; 27:5; 30; 22:5/
0º
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
5; D
2
4x1
x2
x3
3
5; D
2
433453
3
5
n nn
ΠA !ΠC !
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
A B m # nA Rm B
S D f 1; 2; 3g R3
S
f ; ; gR2
R5
Rm " f ; gR2
Rn
1 2 3 R52
1 3 1 2
f 1; 2; 3g
A 6 # 5 7! AR5 R6
A m # n m7! A
a bax D b
a b
.x; y; ´/
ax C by C c´ D d
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
#56
$
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
"21
3
5 D
2
4000
3
5
a
'#1a
$;
#a
a C 2
$(
a; : : : ; f
2
4a00
3
5
2
4bc0
3
5
2
4def
3
5
2
664
a100
3
775
2
664
bc10
3
775
2
664
def1
3
775
A
A D
2
664
1 0 0 02 5 0 03 6 8 04 7 9 10
3
775
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
A 3 # 37! A R3 R3
Rn Rn
R2
#a "bb a
$; a2 C b2 D 1
a b
#43
$ #50
$
(4, 3)
(5, 0)x1
x2
R2
R3
a b
2
4a 0 "b0 1 0b 0 a
3
5
2
4234
3
5 D
2
42p
530
3
5 ; a2 C b2 D 1
x1 x2
x3
x1
2
4378
3
5
WEB
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