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 .
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.
∴{} 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.
For example, ∈ is (1,0,0) and ∈ is (0,0,1)
proof for {} is a basis for
i)Linearly independent
∴
∴{} is linearly independent
ii) Span
For any∈, ∃ such that
=
∴{} spans
∴{} is linearly independent and spans so it is a basis for .