[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.

[Math][Linear Algebra] Field

Though my major is Math, there had been no post about it except some programs related about it.

This post is about field, studied in last semester, Linear Algebra class.

Definition of field:

If nonempty set F satisfies following things, we call F is a field.


[about +]
1. There exist an operation + such that + is closed and well defined under set F
2. + is associative: (a+b)+c=a+(b+c) for all a,b,c in F
3. there exist 0 in F such that a+0=0+a=a for all a in F
(0 is additive identity of F)
4. For all a in F, there exist -a in F such that a+(-a)=(-a)+a=0
   (-a is additive inverse of a)
5.+ is commutative: a+b=b+a for all a,b in F

[about ×]

6. There exist an operation × such that × is closed and well defined under set F

7. × is associative: (a×b)×c=a×(b×c) for all a,b,c in F

8. there exist 1 in F such that a×1=1×a=a for all a in F. 1 is not 0

(1 is multiplicative identity of F)

9. For all nonzero a in F, there exist a^-1 in F such that a+(a^-1)=(a^-1)+a=0 

   (a^-1 is additive inverse of a)

10.× is commutative: a×b=b×a for all a,b in F

11. × is distributive on the left and right : (a+b)×c=a×c+b×c . a×(b+c)=a×b+a×c for all a,b,c in F

N is not a filed since additive inverse does not exist for all element in F.

(for example, -1 is not in F)

Z is not a filed since multiplicative inverse does not exist for all element in F.

(for example, 1/2 is not in F)

Q is a field.
R is a field, too.

Z_2={0,1}is field and the smallest.
We can easily show that Z_2 is a field by showing Z_2 satisfies 1~11.
Z_2 is the smallest field since a field must have at least 2 elements; 0 and nonzero element 1.

The book which I studied last semester.
It is easy to understand and conciseness.












My professor who taught vector calculus recommended to read this book last year.