[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.

[Math][Linear Algebra] Vector space

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.
      















I learned definition of vector is has magnitude and direction in high school.

I learned more mathematical definition of it. Vector is a element of vector space.

Here is the definion:

The set V is a vector space over a field F if if satisfies 1~9

1. ∃+ : VxV→V such that + is closed under V

2. + is associative: (u+v)+w = u+(v+w) for for ∀u,v,w ∈V

3. V has additive identity, 0. u+0=0+u=u for ∀u∈V

4. For each u∈V, ∃-u, additive inverse, exist such that u+(-u)=(-u)+u=0

5. + is commutative. u+v=v+u for ∀u,v ∈V

1~5 is similar to definition of a field: http://sailingkyle.blogspot.com/2011/07/mathlinear-algebra-field.html

6. If a∈F, v∈V then a∙v=av∈V
(V is close under scalar multiplication over a field F)

7. V has associative property of scalar multiplication.
a(bu)=(ab)u for ∀a,b ∈F  ∀u∈V

8. V has distributive property.
a(u+v)=au+av  for ∀a∈F  ∀u,v∈V
(a+b)u= au+bu for ∀a,b ∈F  ∀u∈V

9. Nonzero multiplicative identity of scalar multiplication 1 exist such that 1∙u=u for ∀u∈V

Example of vector space is R^n . Very familiar space.

Though 1~9 looks trivial, it is important.