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Chapter 1: Q21Q (page 1)

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 \(T\).

Short Answer

Expert verified

\(\left( {\begin{aligned}{*{20}{c}}1&0&0\\0&{ - 1}&0\\0&0&1\end{aligned}} \right)\)

Step by step solution

01

Find the order of matrix \(A\)

Use the equation \(T = A{\bf{x}}\). As the order of matrix \(T\) is \(3 \times 1\) and the order of \({\bf{x}}\) is \(3 \times 1\), the order of \(A\) must be \(3 \times 3\).

\(\left( {\begin{aligned}{*{20}{c}}?&?&?\\?&?&?\\?&?&?\end{aligned}} \right)\left( {\begin{aligned}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{aligned}} \right) = \left( {\begin{aligned}{*{20}{c}}{{x_1}}\\{ - {x_2}}\\{{x_3}}\end{aligned}} \right)\)

02

Compare the rows of the matrix

Compare both sides of the equation \(\left( {\begin{aligned}{*{20}{c}}?&?&?\\?&?&?\\?&?&?\end{aligned}} \right)\left( {\begin{aligned}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{aligned}} \right) = \left( {\begin{aligned}{*{20}{c}}{{x_1}}\\{ - {x_2}}\\{{x_3}}\end{aligned}} \right)\) to get the first row of the matrix with unknown elements as \(\left( {\begin{aligned}{*{20}{c}}1&0&0\end{aligned}} \right)\).

03

Compare the rows of the matrix

Compare both sides of the equation \(\left( {\begin{aligned}{*{20}{c}}?&?&?\\?&?&?\\?&?&?\end{aligned}} \right)\left( {\begin{aligned}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{aligned}} \right) = \left( {\begin{aligned}{*{20}{c}}{{x_1}}\\{ - {x_2}}\\{{x_3}}\end{aligned}} \right)\) to get the second row of the matrix with unknown elements as \(\left( {\begin{aligned}{*{20}{c}}0&{ - 1}&0\end{aligned}} \right)\).

04

Compare the rows of the matrix

Compare both sides of the equation \(\left( {\begin{aligned}{*{20}{c}}?&?&?\\?&?&?\\?&?&?\end{aligned}} \right)\left( {\begin{aligned}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{aligned}} \right) = \left( {\begin{aligned}{*{20}{c}}{{x_1}}\\{ - {x_2}}\\{{x_3}}\end{aligned}} \right)\) to get the third row of the matrix with unknown elements as \(\left( {\begin{aligned}{*{20}{c}}0&0&1\end{aligned}} \right)\).

So, the unknown matrix in the equation is \(\left( {\begin{aligned}{*{20}{c}}1&0&0\\0&{ - 1}&0\\0&0&1\end{aligned}} \right)\).

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Most popular questions from this chapter

Consider a dynamical systemwith two components. The accompanying sketch shows the initial state vectorx鈫�0and two eigen vectors蠀1鈫�鈥�鈥�and鈥�鈥�蠀2鈫�of A (with eigen values 位1鈫�and位2鈫�respectively). For the given values of位1鈫�and位2鈫�, draw a rough trajectory. Consider the future and the past of the system.

位1鈫�=0.9,位2鈫�=0.9

Suppose Tand Ssatisfy the invertibility equations (1) and (2), where T is a linear transformation. Show directly that Sis a linear transformation. (Hint: Given u, v in \({\mathbb{R}^n}\), let \({\mathop{\rm x}\nolimits} = S\left( {\mathop{\rm u}\nolimits} \right),{\mathop{\rm y}\nolimits} = S\left( {\mathop{\rm v}\nolimits} \right)\). Then \(T\left( {\mathop{\rm x}\nolimits} \right) = {\mathop{\rm u}\nolimits} \), \(T\left( {\mathop{\rm y}\nolimits} \right) = {\mathop{\rm v}\nolimits} \). Why? Apply Sto both sides of the equation \(T\left( {\mathop{\rm x}\nolimits} \right) + T\left( {\mathop{\rm y}\nolimits} \right) = T\left( {{\mathop{\rm x}\nolimits} + y} \right)\). Also, consider \(T\left( {cx} \right) = cT\left( x \right)\).)

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

11. \(\left[ {\begin{array}{*{20}{c}}3&{ - 4}&2&0\\{ - 9}&{12}&{ - 6}&0\\{ - 6}&8&{ - 4}&0\end{array}} \right]\).

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

Suppose Ais an \(n \times n\) matrix with the property that the equation \(A{\mathop{\rm x}\nolimits} = 0\) has at least one solution for each b in \({\mathbb{R}^n}\). Without using Theorem 5 or 8, explain why each equation Ax = b has in fact exactly one solution.
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