Math 2406: Abstract Vector Spaces

Lecture 5 Plan  (Tuesday, September 4, 2007).

Apostol 3.2-3.6.

Today we look at more examples of what are and are not linear spaces. For spaces which are not, the closure axioms are the most commonly violated. Just as a linear space is an abstraction of a vector space, we can abstract the notion of a subspace of a vector space to define a subspace of a linear space.  One important kind of subspace is the subspace spanned by a set of elements.  Having done this, we can consider subspaces not only of vectors, but of functions. This is important in Fourier analysis, for example.

Review questions:

1. Can you think of a set which fails to be a linear space by failing axioms other than the closure axioms in section 3.2?
2. Is the set of vectors parallel to a given vector N in R^n a linear space?
3. Is the set of vectors in R^n whose components add up to an integer a subspace of the set of all vectors in R^n?