[Math][Linear Algebra] Basis.

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 already know
linearly independent: http://sailingkyle.blogspot.com/2011/08/mathlinear-algebra-linearly-independent.html 
and span: http://sailingkyle.blogspot.com/2011/08/mathlinear-algebra-span.html

Then now I ready for define basis.

Definition of basis
Let V be a vector space over V. For {} ⊆V, if {} is
LINEARLY INDEPENDENT and SPANS V,
then we call{} is a BASIS for V.



Basis can be defined by other ways. Here is another definition.
Alternative definition of basis
If for∀v∈V, if v can be  written as a LINEAR COMBINATION 
of   UNIQUELY,then we call{} is a BASIS for V.


Proof for equivalence.

For each v∈V, if v= for unique .
∴{} spans V.

The solution of =0 is unique since 0 can be written as a linear combination of  uniquely.
Trivial solution of it is  and it is a unique solution.
∴{}is linearly independent. Q.E.D.


Here is a example of basis for a vector space  which called a standard basis.

Def)  i th element is 1, others are all zero.∈,
For example, ∈ is (1,0,0) and ∈ is (0,0,1)

We call {}  is a standard basis. for 

proof for {}  is  a basis for 

i)Linearly independent

If =0,  then=0

∴ 

∴{}  is linearly independent

ii) Span
For any∈,  ∃ such that
=
∴{}  spans 

∴{}  is linearly independent and spans  so it is a basis for .

[Math][Linear Algebra] Span

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.

Span is as important as linear combination. Both linearly independent(I'll post it later)and it are very important to define basis later.

Definition of the Span

The set of linear combination(See http://sailingkyle.blogspot.com/2011/08/mathlinear-algebra-linear-combination.html) of  is called the span of 

denoted by sp{}.

We call sp{} is a spanning set for S=sp{} and using as a verb,

 span S.

Here is useful theorem of span

Theorem 1) Let V is a vector space over a field F. ∈V

If v ∈sp{} then sp{, v} =sp{}

pf) Since  v ∈sp{} ∃ ∈ F such that

v=

if w∈sp{, v} , ∃, b ∈ F such that

w=+bv

    =+b( )

   = 

since F is closed under addition,   ∈F   

∴w ∈sp{}

∴ sp{, v}  ⊆sp{} ---*

if w∈sp{} , ∃∈ F such that

w=

   = + 0 v
∴w ∈sp{, v}
∴ sp{}  ⊆sp{,v} ---**

by * and **,  sp{}  = sp{,v} QED.