Definition of Vector Space
Definition:
A real vector space V is a set of elements together with two operations, addition
and scalar multiplication, satisfying the following properties:
Let u, v, and w be vectors in V, and let c and d be scalars.
Addition:
(  ) u  v is in V.
u v  vu .
u  (v  w)  (u  v)  w .
1.
2.
3. V has a zero vector 0 such that for every u in V, u  0  0  u  u .
4. For every u in V, there is an element  u in V, u  (u )  0 .
Scalar multiplication:
(  ) cu is in V.
5. c(du )  (cd )u .
6. (c  d )u  cu  du .
7. c(u  v)  cu  cv .
8.
1 u  u .
Example:
V1  R n together with standard vector addition and scalar multiplication. Then, V1
is a vector space since for any two vectors u and v in R n and scalar c, both u  v
and cu are in R n . Therefore, conditions (  ) and (  ) are satisfied. In addition,
the conditions (1) to (8) are satisfied (see the previous subsection).
Example:
V2  the set consisting of all m n matrices together with standard matrix addition
and scalar multiplication. V2 is a vector space since for any two m n matrices u
and v and scalar c, both u  v and cu are m n matrices. That is, both u  v
and cu are still in V2 . Therefore, conditions (  ) and (  ) are satisfied. In
addition, the conditions (1) to (8) are satisfied (see section 2).
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Example:
V3  the set consisting of all polynomials of degree 2 or less with the form together
with standard polynomial addition and scalar multiplication. Is V3 a vector space?
We need to examine whether the conditions (  ), (  ), and the conditions (1) to (8)
are satisfied. Let
u  a 2 x 2  a1 x  a 0 ,
v  b2 x 2  b1 x  b0 ,
w  c2 x 2  c1 x  c0
and let c and d be scalars. Then,
Addition:
(  ):
u  v  (a2 x 2  a1 x  a0 )  (b2 x 2  b1 x  b0 )
 (a2  b2 ) x 2  (a1  b1 ) x  (a0  b0 )  V3
since u  v is a polynomial of degree 2 or less.
(1):
u  v  (a2 x 2  a1 x  a0 )  (b2 x 2  b1 x  b0 )
 (a2  b2 ) x 2  (a1  b1 ) x  (a0  b0 )
 (b2  a2 ) x 2  (b1  a1 ) x  (b0  a0 )
 (b2 x 2  b1 x  b0 )  (a2 x 2  a1 x  a0 )  v  u
(2):
u  (v  w)  (a2 x 2  a1 x  a0 )  [(b2  c2 ) x 2  (b1  c1 ) x  (b0  c0 )]
 (a2  b2  c2 ) x 2  (a1  b1  c1 ) x  (a0  b0  c0 )
 [( a2  b2 ) x 2  (a1  b1 ) x  (a0  b0 )]  c2 x 2  c1 x  c0
 (u  v)  w
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(3):
Let 0  0 x 2  0 x  0 . Then,
u  0  (a2  0) x 2  (a1  0) x  (a0  0)  (0  a2 ) x 2  (0  a1 ) x  (0  a0 )  0  u
 a2 x 2  a1 x  a0  u
(4):
Let  u  (a 2 ) x 2  (a1 ) x  (a 0 ). Then,
u  u  [a 2  (a 2 )] x 2  [a1  (a1 )] x  [a0  (a0 )]  0 x 2  0 x  0  0
Scalar multiplication:
(  ):
cu  (ca 2 ) x 2  (ca1 ) x  (ca 0 )  V3
since cu is a polynomial of degree 2 or less.
(5):
c(u  v)  [c(a2  b2 )]x 2  [c(a1  b1 )]x  [c(a0  b0 )]
 (ca2  cb2 ) x 2  (ca1  cb1 ) x  (ca0  cb0 )  cu  cv
(6):
(c  d )u  [(c  d )a2 ]x 2  [(c  d )a1 ]x  [(c  d )a0 ]
 [(ca2 ) x 2  (ca1 ) x  (ca0 )]  [( da2 ) x 2  (da1 ) x  (da0 )]  cu  du
(7):
c(du)  c[( da2 ) x 2  (da1 ) x  (da0 )]  c(da2 ) x 2  c(da1 ) x  c(da0 )
 (cd )a2 x 2  (cd )a1 x  (cd )a0  (cd )u
(8):
1u  1a 2 x 2  1a1 x  1a0  a 2 x 2  a1 x  a0  u
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Note:
Pn  the set consisting of all polynomials of degree n or less with the form
together with standard polynomial addition and scalar multiplication. Then, Pn
is a vector space. In addition, P  the set consisting of all polynomials with the
form together with standard polynomial addition and scalar multiplication. Then, P
is also a vector space.
Example:
V4  the set consisting of all real-valued continuous functions defined on the entire
real line together with standard addition and scalar multiplication. Is V4 a vector
space?
We need to examine whether the conditions (  ), (  ), and the conditions (1) to (8)
are satisfied. Let
u  f (x) ,
v  g (x) ,
w  h(x)
and let c and d be scalars. Then,
Addition:
(  ):
u  v  f ( x)  g ( x) V4
since f ( x)  g ( x) is still a continuous function.
(1):
u  v  f ( x)  g ( x)  g ( x)  f ( x)  v  u
(2):
u  (v  w)  f ( x)  g ( x)  h( x)   f ( x)  g ( x)  h( x)
 (u  v)  w
(3):
Let the zero vector
0  0 . Then,
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u  0  f ( x)  0  f ( x)  u
(4):
Let
 u   f (x) . Then,
u  (u)  f ( x)   f ( x)  f ( x)  f ( x)  0
Scalar Multiplication:
(  ):
cu  cf ( x) V4
since cf (x) is still a continuous function.
(5):
c(u  v)  c f ( x)  g ( x)  cf ( x)  cg ( x)  cu  cv
(6):
(c  d )u  (c  d ) f ( x)  cf ( x)  df ( x)  cu  du
(7):
c(du)  cdf ( x)  cdf ( x)  (cd ) f ( x)  (cd )u
(8):
1(u)  1 f ( x)  f ( x)  u
Note:
Let V4*  the set of all differentiable functions defined on the entire real line and
V4**  the set of all integrable functions defined on the entire real line. Both V4*
and V4** are vector space under standard addition and scalar multiplication.
Example:
V5  the set consisting of all integers with standard addition and scalar
multiplication. Is V5 a vector space?
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V5 is not a real vector space since for u  1  V5 , and c  0.7 ,
cu  0.7  1  0.7  V5 ,
condition (  ) is not satisfied.
Example:
V6  the set consisting of all vectors in R 2 with standard addition and nonstandard
scalar multiplication defined by
 x1  cx1 
c    
 x2   0 
. Is V 6 a vector space?
c 
V 6 is not a real vector space since for u   1   R 2 , c1 , c2  R, c2  0 ,
c2 
 c  c   c 
1 u  1 1    1    1   u ,
c2   0  c2 
condition (8) is not satisfied.
Example:
V7  the set consisting of only second degree polynomials with standard addition
and scalar multiplication. Is V 7 a vector space?
V 7 is not a real vector space since for
u  x2
and
v  x2
u  v  x 2  ( x 2 )  0  V7 ,
condition (  ) is not satisfied.
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Example:
V8  the set consisting of all real-valued continuous functions such that f (1)  3 .
Suppose the operations are standard addition and scalar multiplication. Is V8 a vector
space?
V8 is not a real vector space since for
u  f ( x) V8 ,
v  g ( x) V8
f (1)  g (1)  3  3  6  3
 u  v  f ( x)  g ( x)  V8
condition (  ) is not satisfied.
Important Result:
Let u be any element of a real vector space V. Then,
(a) 0u  0.
(b) c0  0, c  R, 0  V .
(c) cu  0.  c  0 or u  0.
(d) (1)u  u.
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,