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Chapter 1: Q14E (page 1)

Let \(T:{\mathbb{R}^2} \to {\mathbb{R}^2}\) be the linear transformation with standard matrix \(A = \left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}\end{array}} \right]\), where \({a_1}\) and \({a_2}\) are shown in the figure. Using the figure, draw the image of \(\left[ {\begin{array}{*{20}{c}}{ - 1}\\3\end{array}} \right]\) under the transformation \(T\).

Short Answer

Expert verified

Solve the equation \(T\left( x \right) = Ax\) using thelinear transformation.

\(\begin{aligned} T\left( x \right) &= Ax\\ &= \left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}\end{array}} \right]x\\ &= {x_1}{a_1} + {x_2}{a_2}\end{aligned}\)

Step by step solution

01

Solve the equation \(T\left( x \right) = Ax\)

Solve the equation \(T\left( x \right) = Ax\) using thelinear transformation.

\(\begin{aligned} T\left( x \right) &= Ax\\ &= \left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}\end{array}} \right]x\\ &= {x_1}{a_1} + {x_2}{a_2}\end{aligned}\)

02

Find the image of \(\left[ {\begin{array}{*{20}{c}}{ - 1}\\3\end{array}} \right]\)

Find the image of \(\left[ {\begin{array}{*{20}{c}}{ - 1}\\3\end{array}} \right]\) using the equation \(T\left( x \right) = {x_1}{a_1} + {x_2}{a_2}\).

\(T\left( x \right) = - {a_1} + 3{a_2}\)

03

Locate the image of \(\left( { - 1,3} \right)\).

In the given graph, locate the image of \(\left( { - 1,3} \right)\) by forming a parallelogram as shown below:

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Most popular questions from this chapter

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 \({{\mathop{\rm v}\nolimits} _1} = \left[ {\begin{array}{*{20}{c}}1\\0\\{ - 2}\end{array}} \right],{v_2} = \left[ {\begin{array}{*{20}{c}}{ - 3}\\1\\8\end{array}} \right],\) and \({\rm{y = }}\left[ {\begin{array}{*{20}{c}}h\\{ - 5}\\{ - 3}\end{array}} \right]\). For what values(s) of \(h\) is \(y\) in the plane generated by \({{\mathop{\rm v}\nolimits} _1}\) and \({{\mathop{\rm v}\nolimits} _2}\)

In Exercises 13 and 14, determine if \({\mathop{\rm b}\nolimits} \) is a linear combination of the vectors formed from the columns of the matrix \(A\).

14. \(A = \left[ {\begin{array}{*{20}{c}}1&{ - 2}&{ - 6}\\0&3&7\\1&{ - 2}&5\end{array}} \right],{\mathop{\rm b}\nolimits} = \left[ {\begin{array}{*{20}{c}}{11}\\{ - 5}\\9\end{array}} \right]\)

Consider a dynamical system x→(t+1)=Ax→(t)with two components. The accompanying sketch shows the initial state vector x→0and two eigenvectors υ1→  and  υ2→of A (with eigen values λ1→andλ2→ respectively). For the given values of λ1→andλ2→, draw a rough trajectory. Consider the future and the past of the system.

λ1→=1.2,λ2→=1.1

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.

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