[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.