Question: 19. Let \(S\) be an affine subset of \({\mathbb{R}^n}\) , suppose \(f:{\mathbb{R}^n} \to {\mathbb{R}^m}\)is a linear transformation, and let \(f\left( S \right)\) denote the set of images \(\left\{ {f\left( {\rm{x}} \right):{\rm{x}} \in S} \right\}\). Prove that \(f\left( S \right)\)is an affine subset of \({\mathbb{R}^m}\).

It is given that \(\left\{ {f\left( {\rm{x}} \right):{\rm{x}} \in S} \right\}\). If \({\bf{p,q}} \in f\left( S \right)\), \({\bf{r,s}} \in S\), \(f\left( {\bf{r}} \right) = {\bf{p}}\) and \(f\left( {\bf{s}} \right) = {\bf{q}}\). Suppose \(t\) be any vector for the subset \(\mathbb{R}\) such that \(t \in \mathbb{R}\).

Let z is in\(\mathbb{R}\)and since\(f:{\mathbb{R}^n} \to {\mathbb{R}^m}\)is a linear transformation Then, the linearity property \(z = \left( {1 - t} \right){\rm{p}} + t{\rm{q}}\) must be in \(f\left( s \right)\).

Writez as shown below:

\(\begin{array}{c}z = \left( {1 - t} \right){\rm{p}} + t{\rm{q}}\\ = \left( {1 - t} \right)f\left( {\rm{r}} \right) + tf\left( {\rm{s}} \right)\\ = f\left( {\left( {1 - t} \right){\rm{r}} + t{\rm{s}}} \right)\end{array}\)

As \(S\) is an affine subset of \({\mathbb{R}^n}\) such that \(\left\{ {f\left( x \right):x \in S} \right\}\),then for \(f\left( {\left( {1 - t} \right)r + ts} \right)\), \(\left( {1 - t} \right)r + ts \in S\).

This implies that \({\bf{z}} \in S\) and \(f\left( S \right)\) is also considered as an affine subset of \({\mathbb{R}^m}\) as z is in \({\mathbb{R}^n}\) and \(f:{\mathbb{R}^n} \to {\mathbb{R}^m}\).