403/503 – Homework 2

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 T:V\to W be a linear map between finite dimensional vector spaces over {\mathbb F}={\mathbb R}. For S\subseteq W, we denote by T^{-1}[S] the preimage of S under T. This is the set of all vectors v\in V such that Tv\in S. 

Recall that a subset C of a vector space U over {\mathbb R} is convex iff whenever u_1,u_2\in C and \alpha\in[0,1], then also \alpha u_1+(1-\alpha)u_2\in C. Show that T^{-1}[S] is convex iff S\cap{\rm ran}(T) is convex. Is it necessary that T is linear for this to hold, or does it suffice that T is continuous?

Assume that S is a subspace of W, and show that T^{-1}[S] is a subspace of V.

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2 Responses to 403/503 – Homework 2

  1. When you ask the question “Is it necessary that T is linear for this to hold, or does it suffice that T 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 V and W. My guess is since they are finite dimensional and isomorphic to \mathbb{R}^n for some n and \mathbb{R}^n is a metric space, we would use that.

  2. Yes, I am thinking of V,W as {\mathbb R}^n,{\mathbb R}^m, and I would like a proof or a counterexample in this context.

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