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.

u
v
u
 v

W
c
u
cu

W
Example:
W
 the subset of
1
3
R consisting of all vectors of the form,



 a
0
0




, a

R
, together with standard addition and scalar multiplication. Is W a subspace of
1
3
R ?
We need to check if the conditions (1) and (2) are satisfied. Let u





 a
1
0
0



,

 v





 a
2
0
0





, c

R
.
Then,
(1): u
 v




 a
1
0
0








 a
0
0
2







 
 a
1
0
0 a
2





W
1 .
1

(2): cu




 ca
0
0
1





W
1 .

W is a subspace of
1
3
R .
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
W
2
 the subset of V consisting of all n
 n diagonal matrices.
Is W
2
a subspace of V?
Let u




 a
11 
 0

0 a
0

0
22




0
0
 a nn







W
2
, v




 b
11 
 0

0 b
0

22
0




0
0
 b nn







W
2
, and c

R .
(1): u
 v



 a
11 


0

0 b
11 since u
 v is still a diagonal matrix.
(2):
0 a
22
 b
22

0



 a nn
0
0

 b nn






W
2 cu



 ca
11 
 0

0
0 ca
22

0




0
0
 ca nn






W
2 since cu is still a diagonal matrix.

W
2
is a subspace of V.
2

Example: .
P n
 the set consisting of all polynomials of degree n or less with the form together with standard polynomial addition and scalar multiplication . P 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,
P is a subspace of P .
Example:
V
4

the set consisting of all real-valued continuous functions defined on the entire real line together with standard addition and scalar multiplication. Let V
4
*  t he set of all differentiable functions defined on the entire real line together with standard addition and scalar multiplication. Then, V
4
is a real vector space. Also.
*
V is a
4 subspace of V
4
.
Example:
W
 the subset of
3
3
R consisting of all vectors of the form,



 a a b
2




, a , b

R
, together with standard addition and scalar multiplication. Is W a subspace of
3
3
R ?
Let u





 a a
1
2 b
1
1





, v





 a
2 a
2
2 b
2



,

 c

R
.
Then,
3

(1): u
 v




 a
1 a
1
2 b
1








 a a
2
2
2 b
2








 a a
1
2 b
1
1


 a
2 a
2
2 b
2








 a
1
 a
1

 a
2 a
2

2 b
1
 b
2





W
3 .
Therefore, u
 v

W
3 .

W is not a subspace of
3
3
R .
Example:
V
3

the set consisting of all polynomials of degree 2 or less with the form together with standard polynomial addition and scalar multiplication. V
3
is a vector space.
Let
W
 the subset of
4
V
3
consisting of all polynomials of the form ax
2  bx
 c , a
 b
 c

2
.
Is W a subspace of
4
V
3
?
Let u
 a
2 x
2  a
1 x
 a
0

W
4 and v
 b
2 x
2  b
1 x
 b
0

W
4
.
Then, a
2
 a
1
 a
0

2
and b
2
 b
1
 b
0

2
. Thus, u
 v


(
( a
2 x
2 a
2

 b
2
) a
1 x x
2


( a
0
)
 a
1
 b
( b
2
1
) x x
2

(
 b
1 x
 b
0
) a
0
 b
0
)

W
4 since
 a
2
 b
2
  a
1
 b
1
  a
0
 b
0


( a
2
 a
1
 a
0
)

 b
2
 b
1
 b
0


2

2

4
.

W is not a subspace of
4
V
3
.
4