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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 \({{\mathop{\rm a}\nolimits} _1} = \left[ {\begin{array}{*{20}{c}}1\\4\\{ - 2}\end{array}} \right],{{\mathop{\rm a}\nolimits} _2} = \left[ {\begin{array}{*{20}{c}}{ - 2}\\{ - 3}\\7\end{array}} \right],\) and \({\rm{b = }}\left[ {\begin{array}{*{20}{c}}4\\1\\h\end{array}} \right]\). For what values(s) of \(h\) is \({\mathop{\rm b}\nolimits} \) in the plane spanned by \({{\mathop{\rm a}\nolimits} _1}\) and \({{\mathop{\rm a}\nolimits} _2}\)?

Determine h and k such that the solution set of the system (i) is empty, (ii) contains a unique solution, and (iii) contains infinitely many solutions.

a. \({x_1} + 3{x_2} = k\)

\(4{x_1} + h{x_2} = 8\)

b. \( - 2{x_1} + h{x_2} = 1\)

\(6{x_1} + k{x_2} = - 2\)

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.

15. \({{\mathop{\rm v}\nolimits} _1} = \left[ {\begin{array}{*{20}{c}}7\\1\\{ - 6}\end{array}} \right],{v_2} = \left[ {\begin{array}{*{20}{c}}{ - 5}\\3\\0\end{array}} \right]\)

Question: Determine whether the statements that follow are true or false, and justify your answer.

16: There exists a 2x2 matrix such that

A[11]=[12]andA[22]=[21].

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

13. \(A = \left[ {\begin{array}{*{20}{c}}1&{ - 4}&2\\0&3&5\\{ - 2}&8&{ - 4}\end{array}} \right],{\mathop{\rm b}\nolimits} = \left[ {\begin{array}{*{20}{c}}3\\{ - 7}\\{ - 3}\end{array}} \right]\)
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