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Chapter 1: Q27E (page 1)

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.

Short Answer

Expert verified
  1. The line through the vectors \(p\) and \(q\) in \(({\mathbb{R}^n}\) can be written in the parametric equation \((x = \left( {1 - t} \right)p + tq\).
  2. \((T\) maps a line segment to a single point \((T\left( p \right)\) for \((T\left( p \right) = T\left( q \right)\) and \((T\) maps a line segment to a line segment if \((T\left( p \right) \ne T\left( q \right)\).

Step by step solution

01

Find the parametric equation for a line through \({\bf{p}}\) and\({\bf{q}}\)

(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 \((x = \left( {1 - t} \right)p + tq\).

02

Find the image of the line segment from  \({\bf{p}}\) to \({\bf{q}}\)

(b)

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

03

Determine the image if \(({\bf{T}}\left( {\bf{p}} \right) = {\bf{T}}\left( {\bf{q}} \right)\)

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 \((T\left( p \right)\).

04

Determine the image if \(({\bf{T}}\left( {\bf{p}} \right) \ne {\bf{T}}\left( {\bf{q}} \right)\)

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.

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Most popular questions from this chapter

In Exercise 2, compute \(u + v\) and \(u - 2v\).

2. \(u = \left[ {\begin{array}{*{20}{c}}3\\2\end{array}} \right]\), \(v = \left[ {\begin{array}{*{20}{c}}2\\{ - 1}\end{array}} \right]\).

Consider the problem of determining whether the following system of equations is consistent for all \({b_1},{b_2},{b_3}\):

\(\begin{aligned}{c}{\bf{2}}{x_1} - {\bf{4}}{x_2} - {\bf{2}}{x_3} = {b_1}\\ - {\bf{5}}{x_1} + {x_2} + {x_3} = {b_2}\\{\bf{7}}{x_1} - {\bf{5}}{x_2} - {\bf{3}}{x_3} = {b_3}\end{aligned}\)

  1. Define appropriate vectors, and restate the problem in terms of Span \(\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3}} \right\}\). Then solve that problem.
  1. Define an appropriate matrix, and restate the problem using the phrase 鈥渃olumns of A.鈥�
  1. Define an appropriate linear transformation T using the matrix in (b), and restate the problem in terms of T.

Let \(A = \left[ {\begin{array}{*{20}{c}}1&0&{ - 4}\\0&3&{ - 2}\\{ - 2}&6&3\end{array}} \right]\) and \(b = \left[ {\begin{array}{*{20}{c}}4\\1\\{ - 4}\end{array}} \right]\). Denote the columns of \(A\) by \({{\mathop{\rm a}\nolimits} _1},{a_2},{a_3}\) and let \(W = {\mathop{\rm Span}\nolimits} \left\{ {{a_1},{a_2},{a_3}} \right\}\).

  1. Is \(b\) in \(\left\{ {{a_1},{a_2},{a_3}} \right\}\)? How many vectors are in \(\left\{ {{a_1},{a_2},{a_3}} \right\}\)?
  2. Is \(b\) in \(W\)? How many vectors are in W.
  3. Show that \({a_1}\) is in W.[Hint: Row operations are unnecessary.]

Find the general solutions of the systems whose augmented matrices are given as

12. \(\left[ {\begin{array}{*{20}{c}}1&{ - 7}&0&6&5\\0&0&1&{ - 2}&{ - 3}\\{ - 1}&7&{ - 4}&2&7\end{array}} \right]\).

Suppose Ais an \(n \times n\) matrix with the property that the equation \(Ax = 0\)has only the trivial solution. Without using the Invertible Matrix Theorem, explain directly why the equation \(Ax = b\) must have a solution for each b in \({\mathbb{R}^n}\).
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