4-1 4.1 Vectors in R n a sequence of n real number ) ( , , 2 , 1 n x x x • An ordered n-tuple: the set of all ordered n-tuple n-space: R n • Notes: (1) An n-tuple can be viewed as a point in R n with the x i ’s as its coordinates. (2) An n-tuple can be viewed as a vector ) , , , ( 2 1 n x x x ) , , , ( 2 1 n x x x ) , , , ( 2 1 n x x x x Chapter 4 Vector Spaces
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4-1 4.1 Vectors in R n a sequence of n real number An ordered n-tuple: the set of all ordered n-tuple n-space: R n Notes: (1) An n-tuple can be viewed.
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4-1
4.1 Vectors in Rn
a sequence of n real number )( ,,2,1 nxxx
• An ordered n-tuple:
the set of all ordered n-tuple
n-space: Rn
• Notes:
(1) An n-tuple can be viewed as a point in Rn w
ith the xi’s as its coordinates.
(2) An n-tuple can be viewed as a vector
in Rn with the xi’s as its components.
),,,( 21 nxxx
),,,( 21 nxxx ),,,( 21 nxxxx
Chapter 4 Vector Spaces
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n = 4
= set of all ordered quadruple of real numbers
R4
= 4-space
),,,( 4321 xxxx
R1
= 1-space = set of all real number
n = 1
n = 2 R2
= 2-space = set of all ordered pair of real numbers ),( 21 xx
n = 3 R3 = 3-space
= set of all ordered triple of real numbers ),,( 321 xxx
• Ex:
a point
21, xx
a vector
21, xx
0,0
4-3
nn vvvuuu ,,, ,,,, 2121 vu
Equal: if and only if vu nn vuvuvu , , , 2211
Vector addition (the sum of u and v): nn vuvuvu , , , 2211 vu
Scalar multiplication (the scalar multiple of u by c): ncucucuc ,,, 21 u
Notes:
The sum of two vectors and the scalar multiple of a vector
in Rn are called the standard operations in Rn.
(two vectors in Rn)
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Negative:
),...,,,( 321 nuuuu u
Difference: ) ,..., , ,( 332211 nn vuvuvuvu vu
Zero vector:)0 ..., ,0 ,0(0
Notes:
(1) The zero vector 0 in Rn is called the additive identity in Rn.
(2) The vector –v is called the additive inverse of v.
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4-6
4-7
Notes:
A vector in can be viewed as:),,,( 21 nuuu u nR
],,,[ 21 nu uu u
nu
u
u
2
1
uor a n×1 column matrix (column vector):
a 1×n row matrix (row vector):
4-8
), , ,(
) , , ,() , , ,(
2211
2121
nn
nn
vuvuvu
vvvuuu
vu
], , ,[
] , , ,[], , ,[
2211
2121
nn
nn
vuvuvu
vvvuuu
vu
nnnn vu
vu
vu
v
v
v
u
u
u
22
11
2
1
2
1
vu
Vector addition Scalar multiplication
nn cu
cu
cu
u
u
u
cc
2
1
2
1
u
), ,,(
),,,(
21
21
n
n
cucucu
u uucc
u
],,,[
],,,[
21
21
n
n
cu cucu
u uucc
u
• The matrix operations of addition and scalar multiplication give the same results as the corresponding vector representations
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4.2 Vector Spaces
• Notes: A vector space consists of four entities:
a set of vectors, a set of scalars, and two operations
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• Examples of vector spaces:
(1) n-tuple space: Rn
),,(),,(),,( 22112121 nnnn vuvuvuvvvuuu
),,(),,( 2121 nn kukukuuuuk
(2) Matrix space: (the set of all m×n matrices with real values)nmMV
Ex: : (m = n = 2)
22222121
12121111
2221
1211
2221
1211
vuvu
vuvu
vv
vv
uu
uu
2221
1211
2221
1211
kuku
kuku
uu
uuk
vector addition
scalar multiplication
vector addition
scalar multiplication
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(3) n-th degree polynomial space: (the set of all real polynomials of degree n or less)
)(xPV n
nnn xbaxbabaxqxp )()()()()( 1100
nn xkaxkakaxkp 10)(
(4) Function space: (the set of all real-valued
continuous functions defined on the entire real line.)
)()())(( xgxfxgf
),( cV
)())(( xkfxkf
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• Notes: To show that a set is not a vector space, you need only to find one axiom that is not satisfied.
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4.3 Subspaces of Vector Space
Trivial subspace:
Every vector space V has at least two subspaces.
(1) Zero vector space {0} is a subspace of V.
(2) V is a subspace of V.
Definition of Subspace of a Vector Space: A nonempty subset W of a vector space V is called a subspace of V if W is itself a vector space under the operations of addition and scalar multiplication defined in V.
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4.4 Spanning Sets and Linear Independence
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4-18
VS
SV
V S
VS
ofset spanning a is
by )(generated spanned is
)(generates spans
)(span
Notes:
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)( (1) SspanS )()(,, (2) 212121 SspanSspanSSVSS
Notes:
,21 SS dependentlinearly is dependent linearly is 21 SS
t independenlinearly is t independenlinearly is 12 SS
Notes:
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dependent.linearly is SS 0• Note:
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4.5 Basis and dimension
Notes:
(1) the standard basis for R3:
{i, j, k} i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1)