Question

Let \(f:{\mathbb{R}^n} \to {\mathbb{R}^m}\) be a linear transformation and let T be a convex subset of \({\mathbb{R}^m}\). Prove that the set \(S = \left\{ {{\bf{x}} \in {\mathbb{R}^n}:f\left( {\bf{x}} \right) \in T} \right\}\) is a convex subset of \({\mathbb{R}^n}\).

Short answer

\(f\left( S \right)\) is a convex subset of \({\mathbb{R}^n}\).

Step-by-step solution

Recall the definition of the set

Since S is a subset of \({\mathbb{R}^n}\) then for, \({\bf{x}},{\bf{y}} \in S\),then by definition of the set

\(f\left( {\bf{x}} \right),f\left( {\bf{y}} \right)\)

Check that \(f\left( S \right)\) is convex

The equation of the line between point x andyis:

\(\left( {1 - t} \right){\bf{x}} + t{\bf{y}}\)

Apply transformation on theequation of the line:

\(\begin{aligned}{}f\left[ {\left( {1 - t} \right){\bf{x}} + t{\bf{y}}} \right] = f\left[ {\left( {1 - t} \right){\bf{x}}} \right]f\left( {t{\bf{y}}} \right)\\ = \left( {1 - t} \right)f\left( {\bf{x}} \right) + tf\left( {\bf{y}} \right)\end{aligned}\)

This is a line segment between points \(f\left( {\bf{x}} \right)\) and \(f\left( {\bf{y}} \right)\). So, that is contained in T.

\(\left( {1 - t} \right)f\left( {\bf{x}} \right) + tf\left( {\bf{y}} \right) \in T\)

So, \(f\left( S \right)\) is a convex subset of \({\mathbb{R}^n}\).