and 
; learn how to determine a basis for a set of vectors;
learn how to determine the coordinates for a vector relative to a basis.
is
, 
, ...,
. This is a basis because any vector in
can be written as a linear combination of the 
's.
The monomials 
form the standard basis for , the set
of all polynomials of degree n in the variable t. A basis can always
be found for a set of n vectors from 
. Just put the vectors (as columns)
into a matrix A and reduce A to echelon form. The pivot
columns of the reduced matrix are a basis for the subspace spanned by
the original set of vectors (also called Col(A)). Given a matrix A, if you
want to find a basis for Nul(A), just take the set described in the previous
lesson that was used to explicitly determine Nul(A); this is already a basis.
Once a basis is available for a subspace, every vector in that subspace has
a unique representation as a linear combination of the basis vectors. The
coordinates of some vector, relative to a basis, are the multipliers
that would be used to express the original vector as a linear combination of
the basis vectors. If you want to find the coordinates for a vector 
in
relative to some basis, put the basis vectors into a matrix A and
solve 
to find the coordinates vector 
. Remember that a
polynomial of degree n can be represented by a vector containing the
n+1 coefficients for the polynomial. Questions about bases and coordinate
representations for polynomials can be answered by working with the vectors
that represent the polynomials.