Learning Outcomes

After completing this chapter, students should be able to do the following.

  • Use axioms for abstract vector spaces (over the real or complex fields) 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.