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 {


LINEARLY INDEPENDENT and SPANS V,
then we call{

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


Proof for equivalence.


∴{

The solution of


Trivial solution of it is

∴{

Here is a example of basis for a vector space




For example,






proof for {


i)Linearly independent

∴

∴{

ii) Span
For any





∴{


∴{


