**Terminology:**- basis, coordinates relative to a basis.
**Objectives:**- understand what a basis is; learn what the standard bases are for 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.
**Reading Assignment:**- Chapter 4, Sections 4.3-4.4 (pages 231-247).
**Lesson Outline****Key Ideas and Discussion:**-
A basis for a subspace is just a minimal set of vectors that can be used (in a
linear combination) to form any vector in the subspace. A set of basis
vectors must be independent, because if they were not, one of them
could be written as a linear
combination of the others; such a vector could then be removed from the
original set to form a smaller spanning set for the subspace.
A basis for a vector space is not unique, so standard bases are often given
for the most important vector spaces. The standard basis for 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. **Practice Problems:**- 4.3.3, 5, 9, 13, 15, 21 (pp. 237-239);
4.4.3, 7, 11, 13, 15, 27 (pp. 248-249).
**Assignment**-

Thu May 28 14:15:28 PDT 1998