[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] Linearly Independent

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.

      

Both linearly independent and span are important for define basis, a useful concept of linear algebra.

We defined span in previous post, see: http://sailingkyle.blogspot.com/2011/08/mathlinear-algebra-span.html

Here is definition of linearly independent.

 Def) Let V is a vector space over a field F.  are elements of V.

 We call  {}is linearly independent iff  the solution of   for is only

If {}is NOT linearly independent, 

we say it is linearly dependent.

Example: {(1,0,0), (0,1,0), (0,1,1) }   ⊆

For a, b,c∈F,  if  a(1,0,0)+b(0,1,0)+c (0,1,1)=0,

(a,b+c,c)=0

∴ a=c=b=0

∴  {(1,0,0), (0,1,0), (0,1,1) } is linearly independent.

Here is a useful theorem.

Thm2)  Let V is a vector space over a field F.  are elements of V. n≥2.

{}is linearly dependent iff ∃ ∈{} , ≠ 0.such that 

 can be written as a linear combination of other vectors.

pf) =>) Since {}is linearly dependent. there exist not all zero scalar  

such that =0

let k be the largest number of 1,2, ... ,n such that  ≠ 0.

k≥2 since if k=1, , which means it is linearly independent.

Then

 is a linear combination of other vectors. Done.

<=)if  ≠ 0 can be written as other vectors,

since all  is not all zero, {} is linearly dependent. Q.E.D.