Download 5.3 Subspace

Subspace
Definition of subspace:
W is called a subspace of a real vector space V if
1. W is a subset of the vector space V.
2. W is a vector space with respect to the operations in V.
Important Result:
W is a subspace of a real vector space V 
1. If u and v are any vectors in W, then
u  v W .
2. If c is any real number and u is any vector in W, then
cu W .
Example:
W1  the subset of R 3 consisting of all vectors of the form,
a 
 0 , a  R
,
 

0 

together with standard addition and scalar multiplication. Is W1 a subspace of R 3 ?
We need to check if the conditions (1) and (2) are satisfied. Let
a1 
a 2 
u   0 , v   0 , c  R
.
 0 
 0 
Then,
(1):
a1  a 2  a1  a 2 
u  v   0    0    0   W1 .
 0   0   0 
1
(2):
ca1 
cu   0   W1 .
 0 
 W1 is a subspace of R 3 .
Example:
Let the real vector space V be the set consisting of all n  n matrices together with
the standard addition and scalar multiplication. Let
W2  the subset of V consisting of all n  n diagonal matrices.
Is W2 a subspace of V?
Let
a11
0
u
 

0
0
a 22

0
0 
 0 
 W2 ,
  

 a nn 
b11 0
0 b
22
v



0
0

0
 0 
 W2 , and c  R .
  

 bnn 

(1):
a11  b11
 0
uv  
 

 0



0
 W
2




 a nn  bnn 

0
a 22  b22

0
0
since u  v is still a diagonal matrix.
(2):
ca11
 0
cu  
 

 0
0 
 0 
 W2

 

 ca nn 

0
ca 22

0
since cu is still a diagonal matrix.
 W2 is a subspace of V.
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Example:.
Pn  the set consisting of all polynomials of degree n or less with the form together
with standard polynomial addition and scalar multiplication. Pn is a vector space.
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. Then,
Pn is a subspace of P .
Example:
V4  the set consisting of all real-valued continuous functions defined on the entire
real line together with standard addition and scalar multiplication. Let V4*  the set of
all differentiable functions defined on the entire real line together with standard
addition and scalar multiplication. Then, V4 is a real vector space. Also. V4* is a
subspace of V4 .
Example:
W3  the subset of R 3 consisting of all vectors of the form,
a 
a 2 , a, b  R
 
,

b 

together with standard addition and scalar multiplication. Is W3 a subspace of R 3 ?
Let
 a1 
 a2 
u  a12 , v  a22 , c  R
.
 b1 
 b2 
Then,
3
(1):
 a1   a2   a1  a2   a1  a2 
2
u  v  a12   a22   a12  a22   a1  a2    W3 .
 b1   b2   b1  b2   b1  b2 
Therefore,
u  v  W3 .
 W3 is not a subspace of R 3 .
Example:
V3  the set consisting of all polynomials of degree 2 or less with the form together
with standard polynomial addition and scalar multiplication. V3 is a vector space.
Let
W4  the subset of V3 consisting of all polynomials of the form
ax 2  bx  c, a  b  c  2 .
Is W4 a subspace of V3 ?
Let
u  a2 x 2  a1 x  a0  W4
and
v  b2 x 2  b1 x  b0  W4 .
Then,
a2  a1  a0  2
and
b2  b1  b0  2 . Thus,
u  v  (a2 x 2  a1 x  a0 )  (b2 x 2  b1 x  b0 )
 (a2  b2 ) x 2  (a1  b1 ) x  (a0  b0 ) W4
since
a2  b2   a1  b1   a0  b0   (a2  a1  a0 )  b2  b1  b0   2  2  4 .
 W4 is not a subspace of V3 .
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