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Chapter 1: Q6E (page 1)

In Exercises 5–8, use the definition ofAx to write the matrix

equation as a vector equation, or vice versa.

6. \(\left[ {\begin{array}{*{20}{c}}7&{ - 3}\\2&1\\9&{ - 6}\\{ - 3}&2\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{ - 2}\\{ - 5}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1\\{ - 9}\\{12}\\{ - 4}\end{array}} \right]\)

Short Answer

Expert verified

The matrix equation as a vector equation is\( - 2 \cdot \left[ {\begin{array}{*{20}{c}}7\\2\\9\\{ - 3}\end{array}} \right] - 5 \cdot \left[ {\begin{array}{*{20}{c}}{ - 3}\\1\\{ - 6}\\2\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1\\{ - 9}\\{12}\\{ - 4}\end{array}} \right]\).

Step by step solution

01

Write the definition of \(A{\bf{x}}\)

It is known that the column of matrix \(A\) is represented as \(\left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{ \cdot \cdot \cdot }&{{a_n}}\end{array}} \right]\), and vector x is represented as \(\left[ {\begin{array}{*{20}{c}}{{x_1}}\\ \vdots \\{{x_n}}\end{array}} \right]\).

According to the definition, the weights in a linear combination of matrix A columns are represented by the entries in vector x.

The matrix equation as a vector equation can be written as shown below:

\(\begin{array}{c}A{\bf{x}} = \left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{ \cdot \cdot \cdot }&{{a_n}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\ \vdots \\{{x_n}}\end{array}} \right]\\b = {x_1}{a_1} + {x_2}{a_2} + \cdots + {x_n}{a_n}\end{array}\)

The number of columns in matrix \(A\) should be equal to the number of entries in vector x so that \(A{\bf{x}}\) can be defined.

02

Write matrix A and vector x

Consider the equation \(\left[ {\begin{array}{*{20}{c}}7&{ - 3}\\2&1\\9&{ - 6}\\{ - 3}&2\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{ - 2}\\{ - 5}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1\\{ - 9}\\{12}\\{ - 4}\end{array}} \right]\).

Here, \(A = \left[ {\begin{array}{*{20}{c}}7&{ - 3}\\2&1\\9&{ - 6}\\{ - 3}&2\end{array}} \right]\), \({\bf{x}} = \left[ {\begin{array}{*{20}{c}}{ - 2}\\{ - 5}\end{array}} \right]\), and \(b = \left[ {\begin{array}{*{20}{c}}1\\{ - 9}\\{12}\\{ - 4}\end{array}} \right]\).

03

Write matrix A columns and vector x entries

Also, it is observed that \({{\bf{a}}_1} = \left[ {\begin{array}{*{20}{c}}7\\2\\9\\{ - 3}\end{array}} \right]\), \({{\bf{a}}_2} = \left[ {\begin{array}{*{20}{c}}{ - 3}\\1\\{ - 6}\\2\end{array}} \right]\), \({x_1} = - 2\), and \({x_2} = - 5\).

04

Use the definition to write the matrix equation as a vector equation

Write the matrix equation \(\left[ {\begin{array}{*{20}{c}}7&{ - 3}\\2&1\\9&{ - 6}\\{ - 3}&2\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{ - 2}\\{ - 5}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1\\{ - 9}\\{12}\\{ - 4}\end{array}} \right]\) as a vector equationby using the definition as shown below:

\(\begin{array}{c}\left[ {\begin{array}{*{20}{c}}7&{ - 3}\\2&1\\9&{ - 6}\\{ - 3}&2\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{ - 2}\\{ - 5}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1\\{ - 9}\\{12}\\{ - 4}\end{array}} \right]\\ - 2 \cdot \left[ {\begin{array}{*{20}{c}}7\\2\\9\\{ - 3}\end{array}} \right] - 5 \cdot \left[ {\begin{array}{*{20}{c}}{ - 3}\\1\\{ - 6}\\2\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1\\{ - 9}\\{12}\\{ - 4}\end{array}} \right]\end{array}\)

Thus, the matrix equation \(\left[ {\begin{array}{*{20}{c}}7&{ - 3}\\2&1\\9&{ - 6}\\{ - 3}&2\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{ - 2}\\{ - 5}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1\\{ - 9}\\{12}\\{ - 4}\end{array}} \right]\)can be written as a vector equation as \( - 2 \cdot \left[ {\begin{array}{*{20}{c}}7\\2\\9\\{ - 3}\end{array}} \right] - 5 \cdot \left[ {\begin{array}{*{20}{c}}{ - 3}\\1\\{ - 6}\\2\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1\\{ - 9}\\{12}\\{ - 4}\end{array}} \right]\).

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Most popular questions from this chapter

In Exercises 23 and 24, key statements from this section are either quoted directly, restated slightly (but still true), or altered in some way that makes them false in some cases. Mark each statement True or False, and justify your answer. (If true, give the approximate location where a similar statement appears, or refer to a definition or theorem. If false, give the location of a statement that has been quoted or used incorrectly, or cite an example that shows the statement is not true in all cases.) Similar true/false questions will appear in many sections of the text.

23.

a. Every elementary row operation is reversible.

b. A \(5 \times 6\)matrix has six rows.

c. The solution set of a linear system involving variables \({x_1},\,{x_2},\,{x_3},........,{x_n}\)is a list of numbers \(\left( {{s_1},\, {s_2},\,{s_3},........,{s_n}} \right)\) that makes each equation in the system a true statement when the values \ ({s_1},\, {s_2},\, {s_3},........,{s_n}\) are substituted for \({x_1},\,{x_2},\,{x_3},........,{x_n}\), respectively.

d. Two fundamental questions about a linear system involve existence and uniqueness.

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 \(T\left( {cx} \right) = cT\left( x \right)\).)

Suppose a linear transformation \(T:{\mathbb{R}^n} \to {\mathbb{R}^n}\) has the property that \(T\left( {\mathop{\rm u}\nolimits} \right) = T\left( {\mathop{\rm v}\nolimits} \right)\) for some pair of distinct vectors u and v in \({\mathbb{R}^n}\). Can Tmap \({\mathbb{R}^n}\) onto \({\mathbb{R}^n}\)? Why or why not?

In Exercises 3 and 4, display the following vectors using arrows

on an \(xy\)-graph: u, v, \( - {\bf{v}}\), \( - 2{\bf{v}}\), u + v , u - v, and u - 2v. Notice that u - vis the vertex of a parallelogram whose other vertices are u, 0, and \( - {\bf{v}}\).

4. u and v as in Exercise 2

In Exercise 19 and 20, choose \(h\) and \(k\) such that the system has

a. no solution

b. unique solution

c. many solutions.

Give separate answers for each part.

19. \(\begin{array}{l}{x_1} + h{x_2} = 2\\4{x_1} + 8{x_2} = k\end{array}\)
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