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Chapter 1: Q18E (page 1)

In Exercises 17-20, show that \(T\) is a linear transformation by finding a matrix that implements the mapping. Note that \({x_1}\), \({x_2}\),鈥︹�︹�� are not vectors but are enteries in vectors

\(T\left( {{x_1},{x_2}} \right) = \left( {2{x_2} - 3{x_1},{x_1} - 4{x_2},0,{x_2}} \right)\)

Short Answer

Expert verified

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

Step by step solution

01

Express \(T\left( x \right)\) in the form of a matrix

Write the linear transformation\(T\left( x \right)\).

\(T\left( x \right) = \left[ {\begin{array}{*{20}{c}}{2{x_2} - 3{x_1}}\\{{x_1} - 4{x_2}}\\0\\{{x_2}}\end{array}} \right]\)

02

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

\(\left[ {\begin{array}{*{20}{c}}{2{x_2} - 3{x_1}}\\{{x_1} - 4{x_2}}\\0\\{{x_2}}\end{array}} \right] = \left[ A \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right]\)

As \(\left[ x \right]\) has only two entries, matrix \(A\) will have two columns and four rows.

03

Compare the rows of the matrix

From the equation \(\left[ {\begin{array}{*{20}{c}}{2{x_2} - 3{x_1}}\\{{x_1} - 4{x_2}}\\0\\{{x_2}}\end{array}} \right] = \left[ A \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right]\), the first row of matrix \(A\) is \(\left[ {\begin{array}{*{20}{c}}{ - 3}&2\end{array}} \right]\).

04

Compare the rows of the matrix

From the equation \(\left[ {\begin{array}{*{20}{c}}{2{x_2} - 3{x_1}}\\{{x_1} - 4{x_2}}\\0\\{{x_2}}\end{array}} \right] = \left[ A \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right]\), the second row of matrix \(A\) is \(\left[ {\begin{array}{*{20}{c}}1&{ - 4}\end{array}} \right]\).

05

Compare the rows of the matrix

From the equation \(\left[ {\begin{array}{*{20}{c}}{2{x_2} - 3{x_1}}\\{{x_1} - 4{x_2}}\\0\\{{x_2}}\end{array}} \right] = \left[ A \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right]\), the third row of matrix \(A\) is \(\left[ {\begin{array}{*{20}{c}}0&0\end{array}} \right]\).

06

Compare the rows of the matrix

From the equation \(\left[ {\begin{array}{*{20}{c}}{2{x_2} - 3{x_1}}\\{{x_1} - 4{x_2}}\\0\\{{x_2}}\end{array}} \right] = \left[ A \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right]\), the third row of matrix \(A\) is \(\left[ {\begin{array}{*{20}{c}}0&1\end{array}} \right]\).

So, the matrix given in the equation is \(\left[ {\begin{array}{*{20}{c}}{ - 3}&2\\1&{ - 4}\\0&0\\0&1\end{array}} \right]\).

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Most popular questions from this chapter

Let \(u = \left[ {\begin{array}{*{20}{c}}2\\{ - 1}\end{array}} \right]\) and \(v = \left[ {\begin{array}{*{20}{c}}2\\1\end{array}} \right]\). Show that \(\left[ {\begin{array}{*{20}{c}}h\\k\end{array}} \right]\) is in Span \(\left\{ {u,v} \right\}\) for all \(h\) and\(k\).

In Exercise 23 and 24, mark each statement True or False. Justify each answer.

23.

a. A homogeneous equation is always consistent.

b. The equation \(Ax = 0\) gives an explicit description of its solution set.

c. The homogeneous equation \(Ax = 0\) has the trivial solution if and only if the equation has at least one free variable.

d. The equation \(x = p + tv\) describes a line through \({\mathop{\rm v}\nolimits} \) parallel to \(p\).

e. The solution set of \(Ax = b\) is the set of all vectors of the form \({\mathop{\rm w}\nolimits} = p + {v_k}\), where \({v_k}\) is any solution of the equation \(Ax = 0\).

In Exercises 19 and 20, find the parametric equation of the line

through a parallel to b.

19. \({\bf{a}} = \left[ {\begin{array}{*{20}{c}}{ - 2}\\0\end{array}} \right]\), \({\bf{b}} = \left[ {\begin{array}{*{20}{c}}{ - 5}\\3\end{array}} \right]\)

In Exercises 29 鈥� 32, (a) does the equation \(A{\mathop{\rm x}\nolimits} = {\mathop{\rm b}\nolimits} \) have a nontrivial solution and (b) does the equation \(Ax = b\) have at least one solution for every possible \({\mathop{\rm b}\nolimits} \)?

30. \(A\) is a \(3 \times 3\) matrix with three pivot positions.

Describe the possible echelon forms of the matrix A. Use the notation of Example 1 in Section 1.2.

a. A is a \({\bf{2}} \times {\bf{3}}\) matrix whose columns span \({\mathbb{R}^{\bf{2}}}\).

b. A is a \({\bf{3}} \times {\bf{3}}\) matrix whose columns span \({\mathbb{R}^{\bf{3}}}\).
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