College Math Teaching

February 10, 2016

Vector subspaces: two examples

Filed under: linear albegra, pedagogy — Tags: — collegemathteaching @ 8:41 pm

I am teaching linear algebra our of the book by Fraleigh and Beauregard. We are on “subspaces” (subsets of R^n for now) and a subspace is defined to be a set of vectors that is closed under both vector addition and scalar multiplication. Here are a couple of examples of non-subspaces:

1. W= \{(x,y)| xy = 0 \} . Now this space IS closed under scalar multiplication, note that this space IS closed under additive inverses. But it is not closed under addition as [x,0] + [0,y]=[x,y] \notin W for x \neq 0, y \neq 0 .

2. (this example is in the book): the vectors \{(n, m) | n, m \in Z \} are closed under vector addition but not under scalar multiplication.

Advertisements

Blog at WordPress.com.