This homework is due Wednesday, February 24.
Solve the following problems from Chapter 3 of the textbook: 2, 4, 5, 7, 9, 10, 26.
In addition, solve the following problem:
Let be a linear map between finite dimensional vector spaces over For we denote by the preimage of under This is the set of all vectors such that
Recall that a subset of a vector space over is convex iff whenever and then also Show that is convex iff is convex. Is it necessary that is linear for this to hold, or does it suffice that is continuous?
Assume that is a subspace of and show that is a subspace of
When you ask the question “Is it necessary that is linear for this to hold, or does it suffice that is continuous?”, did you want simply our opinion or did you want a proof/counterexample. If a proof/counterexample is desired, can we clarify what topology we have on and . My guess is since they are finite dimensional and isomorphic to for some and is a metric space, we would use that.
Yes, I am thinking of as and I would like a proof or a counterexample in this context.