Intro
Translations in the plane
Examples
HowTo: Gaussian Reduction2.2. Bases.
The following definition is central to this topic.
Definition 2.15. A set 
is called a basis (for the vector space 
) if 
is a linearly
independent spanning set. In other words, 
and 
is linearly
independent.
The following property of bases is fundamental.
Proposition 2.16. Let
be a basis for 
. Then every 
can be written
as a linear combination of the 
in precisely one way.
Proof.Since
is a spanning set, every 
can be written as a linear combination
of elements in 
. We must show that this linear combination is unique. Suppose
that
Then we have
Linear independence of 
means that 
for all 
. □
Example 2.17. The set 
consisting
of elementary vectors in 
is called the canonical basis of 
.
Example 2.18. The set 
for 
, 
consisting
of elementary matrices in 
is called
the canonical basis of .
Proposition 2.19. If 
is a linearly
independent subset of 
, it is a basis
for 
.
Proof.By definition 
is a spanning set for . Since it is linearly independent,
it is a basis for .□
There are two fundamental properties of vector spaces: there is always a basis and any two bases have the same cardinality. In particular, if a vector space is finite-dimensional (so that it has a finite spanning set), any two bases have the same number of elements. This will follow from the following two important results.
