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Chapter 4 Euclidean Vector
Spaces4.1 Euclidean n-Space4.2 Linear Transformations from Rn to Rm
4.3 Properties of Linear Transformations Rn to Rm
4.1 Euclidean n-Space1. Definisi, operasi vektor, dan sifat vektor di ruang-n Euclid2. Hasil kali dalam Euclid
a) Panjang/norm dan jarakb) Sifat panjang dan jarakc) Ketaksamaan Cauchy-Scwartzd) Keortogonalan
3. Alternatif notasi untuk vektor4. Formula matriks untuk hasil kali titik
DefinitionVectors in n-Space
O If n is a positive integer, then an ordered n-tuple is a sequence of n real numbers .
O The set of all ordered n-tuple is called n-space and is denoted by Rn
Definition O 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
),...,, 2211 nn vuvuv(u vu
),...,,( 21 nkukukuk u
O The operations of addition and scalar multiplication in this definition are called the standard operations on Rn.
O The Zero vector in Rn is denoted by 0 and is defined to be the vector 0=(0,0,…,0)
O 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)
O 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
O 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
OIf 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
OThe 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
O 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
Norm and Distance in Euclidean n-Space
O We define the Euclidean norm (or Euclidean length) of a vector u=(u1 ,u2 ,…,un) in Rn by
O 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 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)
Example 3Finding Norm and Distance
O 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
OIf u=(u1 ,u2 ,…,un) and v=(v1 , v2 ,…,vn) are vectors in Rn, then
vuvu
Theorem 4.1.4Properties of Length in Rn
O 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
O 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
OIf u, v, and w are vectors in Rn with the Euclidean inner product, then
22
4
1
4
1vuvuvu
Definition Orthogonality
O 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)
T
If is a matrix, then it follows form
formula (7) and properties of the transpose
that
A (A ) ( A) ( A
( ( (
T T T T
T T T T T T
A n n
A
A A A A A
A
u v v u v u v) u u v
u v v) u v )u v u) u v
u 8
9
T
T
A ( )
A A ( )
v u v
u v u v
Example 5Verifying That
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
System Dot Product Form
085
5472
143
321
321
321
xx x
xxx
xxx
0
5
1
),,()8,5,1(
),,()4,7,2(
),,()1,4,3(
321
321
321
xxx
xxx
xxx
