Spanning Sets and Bases of Vector Spaces (Algebra 2: Lecture 22 Video 1)

Lecture 22: We started this lecture by reviewing some differences in terminology when talking about R-modules and talking about vector spaces.  We then proved that if a set of vectors spans a vector space, and no proper subset of it is a spanning set, then it is a basis.  We saw that this implied that any finitely generated vector space over a field F is a free F-module and also that any finite spanning set contains a basis.  We discussed an example involving F[x]/(f(x)).  We stated the 'Replacement Theorem' about bases and linearly independent sets of vectors.  We saw that if V has a basis of size n, then any spanning set for V has at least n vectors, any set of linearly independent vectors has at most n vectors, which taken together imply that any basis has size n.  This led us to give a careful definition of the dimension of a vector space.  We proved the 'Building Up Lemma', which said that if A is a linearly independent set of vectors in V, then there is a basis of V containing A.  We then proved the 'Replacement Theorem'.  Finally, we proved that if V is an n-dimensional vector space, then V is isomorphic to F^n.  In several places we pointed out that the statements we were proving for vector spaces do not hold more generally for R-modules.

Reading: We very closely followed the presentation of the beginning of Section 11.1 (pages 408-411) in this lecture.