assuming a vector space
, let
be a finite, linearly independant and spanning set of
which we call a basis, each vector in a basis is called a basis vector
let
be a basis of
:
since
is linearly independant then:
this is necessarily true because
has to be 0 for
to zero out and if it wasnt zero then the output vector would have a non-zero in it and it wouldnt be
anymore
a basis doesnt have to be of this simplified form, we can take the basis
, apply elementary row operations to its vectors and we would get another basis for the given vector space
every finitely generated vector space
has atleast one basis
since
is finitely generated there exists a finite set
such that
according to [BROKEN LINK: can_drop_vector_keep_span], there exists
where
is linearly independant and
, therefore
is a basis
is linearly independant
is a basis of
given the vector space
, we find a basis of it:
so
is a basis of
and
, see dimension
consider the vector space
which represents all the matrices with dimensions
over the field
what we are basically looking for is a set of matrices that is linearly independant so that no matrix is a linear combination of the others, which is accomplished with:
and:
we prove that
is linearly independant:
it must be that
and so
is linearly independant because the only linear combination that gives us
is where the coefficients all are 0
given
is a finite set and
and
, then
is linearly dependant
in other words, if we have a finitely generated vector space, if
is a subset of that space such that the number of elements in
is bigger than the number of elements in
, then
is linearly dependant
let
and
such that
let
denote the coefficients, for every
where
there exist the coefficients:
let
be a finite set, if
and
is linearly independant then
let
be bases of
then
is a spanning set and
and
is linearly independant then
is a spanning set and
and
is linearly independant then
therefore
basis of the cartesian coordinate system
the of the cartesian coordinate system consists of the unit vectors that lie along the x, y, and z axes
the x unit vector is denoted by
, the y unit vector by
, and the z unit vector by
we can write any vector in terms of the base vectors