Question: 16. Let \({\rm{v}} \in {\mathbb{R}^n}\)and let \(k \in \mathbb{R}\). Prove that \(S = \left\{ {{\rm{x}} \in {\mathbb{R}^n}:{\rm{x}} \cdot {\rm{v}} = k} \right\}\)is an affine subset of \({\mathbb{R}^n}\).

Assume \({\bf{u,v}} \in S\) and \(t \in S\), which means that \({\bf{u,v}}\) are the vector for the basis \(S\) and that \(t\) are the vector for the basis \(\mathbb{R}\).

Apply the dot property with the vectors formed by a vector \(v\) as shown below:

\(\begin{array}{c}\left( {\left( {1 - t} \right){\bf{u}} + t{\bf{v}}} \right).{\bf{v}} = \left( {1 - t} \right)\left( {{\bf{u}} \cdot {\bf{v}}} \right) + t\left( {{\bf{v}} \cdot {\bf{v}}} \right) = \left( {1 - t} \right)k + tk\\ = k\end{array}\)

According to the definition of \(S\), \(\left( {\left( {1 - t} \right){\bf{u}} + t{\bf{v}}} \right) \in S\). So, \(S\) must be affine set.

Therefore,\(S = \left\{ {{\rm{x}} \in {\mathbb{R}^n}:{\rm{x}} \cdot {\rm{v}} = k} \right\}\)is an affine subset of \({\mathbb{R}^n}\).