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Chapter 1: Q25E (page 1)

Given \({\bf{v}} \ne 0\) and \({\bf{p}}\) in\({\mathbb{R}^{\bf{n}}}\), the line through \({\bf{p}}\) in the direction of \({\bf{v}}\) has the parametric equation\({\bf{x}} = {\bf{p}} + {\bf{tv}}\). Show that a linear transformation \({\bf{T}}:{\mathbb{R}^{\bf{n}}} \to {\mathbb{R}^{\bf{n}}}\) maps this line onto another line or onto a single point (a degenerate line).

Short Answer

Expert verified

If \(T\left( v \right) = 0\) then the image is a single point \(T\left( p \right)\). If \(T\left( v \right) \ne 0\) then the image is a line through \(T\left( p \right)\) in the direction \(T\left( v \right)\).

Step by step solution

01

Find the image of the given parametric equation using the properties of a linear transformation

\(\begin{aligned} T\left( x \right) &= T\left( {p + tv} \right)\\ &= T\left( p \right) + T\left( {tv} \right)\\T\left( x \right) &= T\left( p \right) + tT\left( v \right)\end{aligned}\)

02

Determine the image if \({\bf{T}}\left( {\bf{v}} \right) = {\bf{0}}\)

Whenever \(T\left( v \right) = 0\), \(T\left( x \right) = T\left( p \right) + 0 = T\left( p \right)\).

This implies \(T\) maps the line \(x = p + tv\) onto a single point \(T\left( p \right)\).

03

Determine the image if \({\bf{T}}\left( {\bf{v}} \right) \ne {\bf{0}}\)

Whenever \(T\left( v \right) \ne 0\), \(T\left( x \right) = T\left( p \right) + tT\left( v \right)\).

That is, \(T\) maps the line \(x = p + tv\) onto another line through the point \(T\left( p \right)\) in the direction \(T\left( v \right)\).

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Most popular questions from this chapter

In Exercises 15 and 16, list five vectors in Span \(\left\{ {{v_1},{v_2}} \right\}\). For each vector, show the weights on \({{\mathop{\rm v}\nolimits} _1}\) and \({{\mathop{\rm v}\nolimits} _2}\) used to generate the vector and list the three entries of the vector. Do not make a sketch.

16. \({{\mathop{\rm v}\nolimits} _1} = \left[ {\begin{array}{*{20}{c}}3\\0\\2\end{array}} \right],{v_2} = \left[ {\begin{array}{*{20}{c}}{ - 2}\\0\\3\end{array}} \right]\)

Find the general solutions of the systems whose augmented matrices are given in Exercises 10.

10. \(\left[ {\begin{array}{*{20}{c}}1&{ - 2}&{ - 1}&3\\3&{ - 6}&{ - 2}&2\end{array}} \right]\)

In Exercises 31, find the elementary row operation that transforms the first matrix into the second, and then find the reverse row operation that transforms the second matrix into the first.

31. \(\left[ {\begin{array}{*{20}{c}}1&{ - 2}&1&0\\0&5&{ - 2}&8\\4&{ - 1}&3&{ - 6}\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}1&{ - 2}&1&0\\0&5&{ - 2}&8\\0&7&{ - 1}&{ - 6}\end{array}} \right]\)

Let \(T:{\mathbb{R}^3} \to {\mathbb{R}^3}\) be the linear transformation that reflects each vector through the plane \({x_{\bf{2}}} = 0\). That is, \(T\left( {{x_1},{x_2},{x_3}} \right) = \left( {{x_1}, - {x_2},{x_3}} \right)\). Find the standard matrix of \(T\).

In Exercise 1, compute \(u + v\) and \(u - 2v\).

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