After completing this chapter, students should be able to do the following.
Objectives
Determine whether sets are closed under addition and/or scalar multiplication.
Determine whether a set of vectors is a basis of a subspace or \(\R^n\text{.}\)
Write the coordinates of a vector with respect to a given basis of \(\R^n\text{.}\)
Identify the dimension of the span of a given set of vectors.
Find a basis for each of the following: the row space, the column space, and the null space of a given matrix.
Prove elementary theorems concerning rank of a matrix and the relationship between rank and nullity.
Use axioms for abstract vector spaces (over the real numbers) to discuss examples (and non-examples) of abstract vector spaces such as subspaces of the space of all polynomials.
Discuss the existence of a basis of an abstract vector space.
Describe coordinates of a vector relative to a given basis.
For a linear transformation between vector spaces, discuss its matrix relative to given bases.
Discuss how the matrix of a linear transformation with respect to a basis changes when the basis is changed.
Discuss the advantages of a change of basis that leads to a simplified matrix and simplified description of a linear map.
