[Math][Linear Algebra] Subspace

Finite Dimensional Linear Algebra: I learned Linear Algebra by this book.

Linear Algebra Done Right: My professor who taught vector space when I was freshman recommended to read this book.
  














Subset of vector space can be vector space or not.

For example, {1}⊆R is not vector space since there is no additive identity in the set.

Some subset can be vector space.

Definiton of Subspace

Let V be vector space over a field F. S is subset of V (S⊆V)

S is subspace of V if it satisfies following things.

1)0∈S  (S contains additive identity)
2)If u∈S and v∈S, then  u+v∈S for ∀u,v ∈S (S is closed under +)
3)If a∈F, u∈S, then au∈S for ∀a∈F and ∀u∈S. (S is closed under scalar multiplication)

Example of subspace: S={(0,0,0) and (1,1,1)}⊆(Z_2)^3       (Z_2)^3 is vector space over a field Z_2
1) 0=(0,0,0)∈S

2)(0,0,0) +(0,0,0)=(0,0,0)
(0,0,0)+(1,1,1)=(1,1,1)
(1,1,1)+(0,0,0) =(1,1,1)
(1,1,1)+(1,1,1)=(0,0,0). It shows that S is closed under addition.

3)0(0,0,0)=(0,0,0)
1(0,0,0)=(0,0,0)
0(1,1,1)=(0,0,0)
1(1,1,1)=(1,1,1)   It shows that S is closed under scalar multiplication.

Hence, S is a subspace of V

Subspace of a vector space is also vector space and operation is same as vector space which concluding it.