Question

3. Closure Properties

(a) Using that vector spaces are closed under scalar multiplication, explain why if any nonzero vector from R2 or R3 is in a vector space V, then an entire line’s worth of vectors are in V.

(b) Why isn’t closure under vector addition enough to make the same statement?

4. Subspaces and Spans: The span of a set of vectors from Rn is always a subspace of Rn. This is relevant to the problems below because the closure properties apply.

(a) Geometrically, what is the span of a single, nonzero vector in R3?

(b) What are all the possibilities for the type of space, geometrically speaking, that two nonzero vectors in R3 span? Draw a picture for each possibility.

(c) What are all the possibilities for the type of space, geometrically speaking, that three nonzero vectors in R3 span?

Answer 1