a. Show that the line through vectors \({\bf{p}}\) and \({\bf{q}}\) in \({\mathbb{R}^{\bf{n}}}\) may be written in the parametric equation\({\bf{x}} = \left( {{\bf{1}} - {\bf{t}}} \right){\bf{p}} + {\bf{tq}}\). Refer the following figure.

b. The line segment from \({\bf{p}}\)to \({\bf{q}}\) is the set of points of the form \(\left( {{\bf{1}} - {\bf{t}}} \right){\bf{p}} + {\bf{tq}}\) for \({\bf{0}} \le {\bf{t}} \le {\bf{1}}\) (as shown in the figure below). Show that a linear transformation \({\bf{T}}\) maps this line segment onto a line segment or onto a single point.

(a)

For any point \(x\) in the line passing through \(p\) and \(q\)in the direction of \(q - p\),the parametric equation can be written as \(x = p + t\left( {q - p} \right)\) for all \(t \in \mathbb{R}\). So,

\(\begin{array}{c}x = p + tq - tp\\ = p\left( {1 - t} \right) + tq\end{array}\)

Hence, the line through the vectors \(p\) and \(q\) in \(({\mathbb{R}^n}\) can be written in the parametric equation \((x = \left( {1 - t} \right)p + tq\).

(b)

\((\begin{array}{c}T\left( {\left( {1 - t} \right)p + tq} \right) = T\left( {\left( {1 - t} \right)p} \right) + T\left( {tq} \right)\\ = \left( {1 - t} \right)T\left( p \right) + tT\left( q \right)\,\,\,\,\,\,\,\,\,\,\,0 \le t \le 1\end{array}\)

Suppose \((T\left( p \right) = T\left( q \right)\) ; then

\((\begin{array}{c}T\left( {\left( {1 - t} \right)p + tq} \right) = T\left( p \right) - tT\left( p \right) + tT\left( p \right)\\ = T\left( p \right)\end{array}\) .

This implies \((T\) maps a line segment to a single point \((T\left( p \right)\).

Suppose \((T\left( p \right) \ne T\left( q \right)\) then \((T\left( {\left( {1 - t} \right)p + tq} \right) = \left( {1 - t} \right)T\left( p \right) + tT\left( q \right)\) for \((0 \le t \le 1\).

Thus, \((T\) maps a line segment to a line segment.