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Chapter 4
Vector Spaces
n
4.1 Vectors in R
• An ordered n-tuple:
a sequence of n real number ( x1, x2, , xn )
n
 n-space: R
the set of all ordered n-tuple
• Notes:
n
(1) An n-tuple ( x1 , x2 ,, xn ) can be viewed as a point in R
with the xi’s as its coordinates.
(2) An n-tuple ( x1 , x2 ,, xn ) can be viewed as a vector
x  ( x1 , x2 ,, xn ) in Rn with the xi’s as its components.
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• Ex:
n=2
n=3
n=1
1
R = 1-space
= set of all real number
2
R = 2-space
= set of all ordered pair of real numbers ( x1 , x2 )
3
R = 3-space
= set of all ordered triple of real numbers ( x1 , x2 , x3 )
n=4
4
R = 4-space
= set of all ordered quadruple of real numbers ( x1 , x2 , x3 , x4 )
x1 , x2 
x1 , x2 
a point
0,0
a vector
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u  u1 , u2 ,, un , v  v1 , v2 ,, vn 

Equal:
u  v if and only if
(two vectors in Rn)
u1  v1 , u2  v2 , , un  vn
 Vector
addition (the sum of u and v):
u  v  u1  v1 , u2  v2 , , un  vn 

Scalar multiplication (the scalar multiple of u by c):
cu  cu1 , cu2 ,, cun 

Notes:
The sum of two vectors and the scalar multiple of a vector
n
in R are called the standard operations in Rn.
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
Negative:
 u  (u1 ,u2 ,u3 ,...,un )

Difference:
u  v  (u1  v1 , u2  v2 , u3  v3 ,..., un  vn )

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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
Notes:
A vector u  (u1 , u2 ,, un ) in R n can be viewed as:
a 1×n row matrix (row vector): u  [u1 , u2 ,, un ]
or a n×1 column matrix (column vector):
u1 
u 
u   2
 
 
u n 
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• The matrix operations of addition and scalar multiplication
give the same results as the corresponding vector representations
Vector addition
Scalar multiplication
u  v  (u1 , u2 , , un )  (v1 , v2 , , vn )
cu  c(u1 , u2 ,, un )
 (u1  v1 , u2  v2 ,  , un  vn )
u  v  [u1 , u2 ,  , un ]  [v1 , v2 , , vn ]
 [u1  v1 , u2  v2 ,  , un  vn ]
u1  v1  u1  v1 
u  v  u  v 
u v   2   2   2 2
      
    

un  vn  un  vn 
 (cu1 , cu2 , , cun )
cu  c[u1 , u2 , , un ]
 [cu1 , cu2 , , cun ]
u1  cu1 
u  cu 
cu  c  2    2 
   
   
un  cun 
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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
(u1 , u2 ,un )  (v1 , v2 ,vn )  (u1  v1 , u2  v2 ,un  vn ) vector addition
k (u1 , u2 ,un )  (ku1 , ku2 , kun )
scalar multiplication
(2) Matrix space: V  M mn (the set of all m×n matrices with real values)
Ex: ：(m = n = 2)
u11 u12  v11 v12   u11  v11 u12  v12 
u u   v v   u  v u  v  vector addition
 21 22   21 22   21 21 22 22 
u11 u12   ku11 ku12 
k



u
u
ku
ku
22 
 21 22   21
scalar multiplication
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(3) n-th degree polynomial space: V  Pn (x)
(the set of all real polynomials of degree n or less)
p( x)  q( x)  (a0  b0 )  (a1  b1 ) x    (an  bn ) x n
kp( x)  ka0  ka1 x    kan x n
(4) Function space: V  c(, ) (the set of all real-valued
continuous functions defined on the entire real line.)
( f  g )( x)  f ( x)  g ( x)
(kf )( x)  kf ( x)
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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
 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.
 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.
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4.4 Spanning Sets and Linear Independence
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 Notes:
span ( S )  V
 S spans (generates ) V
V is spanned (generated ) by S
S is a spanning set of V
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 Notes:
(1) S  span( S )
(2) S1 , S2  V , S1  S2  span(S1 )  span(S2 )

Notes: S1  S2 , S1 is linearly dependent  S2 is linearly dependent
S2 is linearly independen t  S1 is linearly independen t
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• Note: 0  S  S is linearly dependent.
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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)
n
(2) the standard basis for R :
{e1, e2, …, en} e1=(1,0,…,0), e2=(0,1,…,0), en=(0,0,…,1)
Ex: R4
{(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1)}
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(3) the standard basis for mn matrix space:
{ Eij | 1im , 1jn }
Ex: 2 2 matrix space:
1 0 0 1 0 0 0 0 
,
,
,






0 0 0 0 1 0 0 1 
(4) the standard basis for polynomials Pn(x):
{1, x, x2, …, xn}
Ex: P3(x)
{1, x, x2, x3}
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• Ex:
(1) Vector space Rn
 basis {e1 , e2 ,  , en}  dim(Rn) = n
(2) Vector space Mmn  basis {Eij | 1im , 1jn}
 dim(Mmn)=mn
(3) Vector space Pn(x)  basis {1, x, x2,  , xn}  dim(Pn(x)) = n+1
(4) Vector space P(x)  basis {1, x, x2, }  dim(P(x)) = 
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4.6 Rank of a Matrix and System of Linear
Equations
 row vectors:
 a11 a12  a1n   A1 
a
 A 
a

a
2  
22
2n 
A   21

 

    


 
A
a
a

a
 m  
m2
mn 
 m1
Row vectors of A
(a11 , a12 ,, a1n )  A(1)
(a21 , a22 ,, a2n )  A(2)

(am1 , am2 ,, amn )  A( m )
 column vectors:
Column vectors of A
 a11 a12  a1n 
a

a

a
22
2n 
A   21
 A1  A2   An 
 

 


a
a

a
m2
mn 
 m1


 a11   a12   a1n 
a   a  a 
 21   22    2 n 
       
    
am1  am 2  amn 
||
||
(1)
(2)
A
A
||
(n)
A
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• Notes: (1) The row space of a matrix is not changed by
elementary row operations.
(2) Elementary row operations can change the
column space.
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 Notes: rank(AT) = rank(A)
Pf:
rank(AT) = dim(RS(AT)) = dim(CS(A)) = rank(A)
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 Notes: (1) The nullspace of A is also called the solution space of
the homogeneous system Ax = 0.
(2) nullity(A) = dim(NS(A))
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•
Notes: (1) rank(A): The number of leading variables in the
solution of Ax=0. (The number of nonzero rows in
the row-echelon form of A)
(2) nullity (A): The number of free variables in the
solution of Ax = 0.
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
Notes: If A is an mn matrix and rank(A) = r, then
Fundamental Space
Dimension
RS(A)=CS(AT)
r
CS(A)=RS(AT)
r
NS(A)
n–r
NS(AT)
m–r
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 Notes: If rank([A|b])=rank(A), then the system Ax=b is consistent.
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4.7 Coordinates and Change of Basis
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• Change of basis problem:
Given the coordinates of a vector relative to one basis B and
want to find the coordinates relative to another basis B'.
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
Ex: (Change of basis) Consider two bases for a vector space V
B  {u1 , u2 }, B  {u1 , u2 }
a 
c 
If [u1 ]B   , [u2 ]B   
b 
d 
i.e., u1  au1  bu 2 , u2  cu1  du 2
 k1 
Let v V , [ v]B   
k 2 
 v  k1u1  k 2u2
 k1 (au1  bu 2 )  k 2 (cu1  du 2 )
 (k1a  k 2 c)u1  (k1b  k 2 d )u 2
 k1a  k2c  a c   k1 
 [ v ]B  


 k 
k
b

k
d
b
d

 2 
2 
 1
 u1 B u2 B  v B
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•
Transition matrix from B' to B:
Let B  {u1 , u 2 ,..., u n } and B  {u1 , u2 ..., un } be two bases
for a vector space V
If [v]B is the coordinate matrix of v relative to B
[v]B‘ is the coordinate matrix of v relative to B'
then [ v]B  P[ v]B
 u1 B , u2 B ,..., un B  vB
where
P  u1 B , u2 B , ..., un B 
is called the transition matrix from B' to B
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
Notes: B  {u1 , u 2 , ..., u n }, B'  {u1 , u2 , ..., un }
vB  [u1 ]B , [u2 ]B , ..., [un ]B  vB  P vB
vB  [u1 ]B , [u 2 ]B , ..., [u n ]B  vB  P 1 vB
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4.8 Applications of Vector Spaces
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