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Chapter 1: Q5E (page 1)

In Exercises 5, write a system of equations that is equivalent to the given vector equation.

5. \({x_1}\left[ {\begin{array}{*{20}{c}}6\\{ - 1}\\5\end{array}} \right] + {x_2}\left[ {\begin{array}{*{20}{c}}{ - 3}\\4\\0\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1\\{ - 7}\\{ - 5}\end{array}} \right]\)

Short Answer

Expert verified

The system of equations is

\(\begin{aligned}{*{20}{c}}{6{x_1} - 3{x_2} = 1}\\{ - {x_1} + 4{x_2} = - 7}\\{\,\,\,\,\,\,\,\,\,\,\,\,\,\,5{x_1} = - 5}\end{aligned}\)

Step by step solution

01

Write the conditions for the vector addition and scalar multiple of a vector by a constant

From the given vector equation, it can be observed that the vectors contain three entries. So, the vectors can be denoted as \({\mathbb{R}^3}\).

Add the corresponding terms ofthe vectors to obtain the sum.

Multiply each entry of a \({\mathbb{R}^3}\) vector by the unknowns \({x_1}\) and \({x_2}\) to obtain the scalar multiple of a vector by a scalar.

02

Compute the scalar multiplication

Consider the vector equations \({x_1}\left[ {\begin{array}{*{20}{c}}6\\{ - 1}\\5\end{array}} \right] + {x_2}\left[ {\begin{array}{*{20}{c}}{ - 3}\\4\\0\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1\\{ - 7}\\{ - 5}\end{array}} \right]\).

Obtain the scalar multiplication of the vector \(\left[ {\begin{array}{*{20}{c}}6\\{ - 1}\\5\end{array}} \right]\) by the unknown \({x_1}\), and obtain the scalar multiplication of the vector \(\left[ {\begin{array}{*{20}{c}}{ - 3}\\4\\0\end{array}} \right]\)by the unknown \({x_2}\).

\(\left[ {\begin{array}{*{20}{c}}{6{x_1}}\\{ - {x_1}}\\{5{x_1}}\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}{ - 3{x_2}}\\{4{x_2}}\\0\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1\\{ - 7}\\{ - 5}\end{array}} \right]\)

03

Add the vectors

Now, add the vectors \(\left[ {\begin{array}{*{20}{c}}{6{x_1}}\\{ - {x_1}}\\{5{x_1}}\end{array}} \right]\) and \(\left[ {\begin{array}{*{20}{c}}{ - 3{x_2}}\\{4{x_2}}\\0\end{array}} \right]\) on the left-hand side of the equation.

\(\left[ {\begin{array}{*{20}{c}}{6{x_1} - 3{x_2}}\\{ - {x_1} + 4{x_2}}\\{5{x_1}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1\\{ - 7}\\{ - 5}\end{array}} \right]\)

04

Equate the vectors and write in the equation form

The unknowns \({x_1}\) and \({x_2}\) must satisfy the system of equations in order to make the equation \(\left[ {\begin{array}{*{20}{c}}{6{x_1} - 3{x_2}}\\{ - {x_1} + 4{x_2}}\\{5{x_1}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1\\{ - 7}\\{ - 5}\end{array}} \right]\)true. So, equate the vectors as shown below:

\(\begin{array}{*{20}{c}}{6{x_1} - 3{x_2} = 1}\\{ - {x_1} + 4{x_2} = - 7}\\{5{x_1} + 0\left( {{x_2}} \right) = - 5}\end{array}\)

Thus, the system of equations is:

\(\begin{aligned}{*{20}{c}}{6{x_1} - 3{x_2} = 1}\\{ - {x_1} + 4{x_2} = - 7}\\{\,\,\,\,\,\,\,\,\,\,\,\,\,\,5{x_1} = - 5}\end{aligned}\)

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Most popular questions from this chapter

Suppose the system below is consistent for all possible values of \(f\) and \(g\). What can you say about the coefficients \(c\) and \(d\)? Justify your answer.

27. \(\begin{array}{l}{x_1} + 3{x_2} = f\\c{x_1} + d{x_2} = g\end{array}\)

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\(k\).

Consider the problem of determining whether the following system of equations is consistent:

\(\begin{aligned}{c}{\bf{4}}{x_1} - {\bf{2}}{x_2} + {\bf{7}}{x_3} = - {\bf{5}}\\{\bf{8}}{x_1} - {\bf{3}}{x_2} + {\bf{10}}{x_3} = - {\bf{3}}\end{aligned}\)

  1. Define appropriate vectors, and restate the problem in terms of linear combinations. Then solve that problem.
  1. Define an appropriate matrix, and restate the problem using the phrase 鈥渃olumns of A.鈥�
  1. Define an appropriate linear transformation T using the matrix in (b), and restate the problem in terms of T.

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

23.

a. Another notation for the vector \(\left[ {\begin{array}{*{20}{c}}{ - 4}\\3\end{array}} \right]\) is \(\left[ {\begin{array}{*{20}{c}}{ - 4}&3\end{array}} \right]\).

b. The points in the plane corresponding to \(\left[ {\begin{array}{*{20}{c}}{ - 2}\\5\end{array}} \right]\) and \(\left[ {\begin{array}{*{20}{c}}{ - 5}\\2\end{array}} \right]\) lie on a line through the origin.

c. An example of a linear combination of vectors \({{\mathop{\rm v}\nolimits} _1}\) and \({{\mathop{\rm v}\nolimits} _2}\) is the vector \(\frac{1}{2}{{\mathop{\rm v}\nolimits} _1}\).

d. The solution set of the linear system whose augmented matrix is \(\left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{{a_3}}&b\end{array}} \right]\) is the same as the solution set of the equation\({{\mathop{\rm x}\nolimits} _1}{a_1} + {x_2}{a_2} + {x_3}{a_3} = b\).

e. The set Span \(\left\{ {u,v} \right\}\) is always visualized as a plane through the origin.

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.

30.\(\left[ {\begin{array}{*{20}{c}}1&3&{ - 4}\\0&{ - 2}&6\\0&{ - 5}&9\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}1&3&{ - 4}\\0&1&{ - 3}\\0&{ - 5}&9\end{array}} \right]\)
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