Week 5 -- Section 4.1

Short-term objectives

  1. Determine whether a given list of vectors is independent.
  2. Given a list of vectors, find an independent set with the same span.
  3. Recognize, and if necessary construct, a basis for a given (low- dimensional) vector space.
  4. Do the foregoing not only in R^n, but in spaces spanned by finite sets of familiar functions (e.g., hyperbolic).

Long-term objectives

  1. Appreciate the value of putting expressions into a normal form, and understand how choosing a basis defines a normal form.
  2. Relate the algebraic definition of "span" to the geometric concept of the smallest plane, line, etc. containing the given set of vectors.
  3. Move back and forth between the various characterizations of linear independence. ("Characterization" means a condition precisely equivalent to the term in question, which might be used as the definition of the term.)