## 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 $V.$

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.