[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.