### Week 5 -- Section 4.1

#### Short-term objectives

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

#### Long-term objectives

- Appreciate the value of putting expressions into a normal form, and
understand how choosing a basis defines a normal form.
- Relate the algebraic definition of "span" to the geometric concept of
the smallest plane, line, etc. containing the given set of vectors.
- 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.)