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Chapter 1: Q6E (page 1)

In Exercises 5 and 6, follow the method of Examples 1 and 2 to write the solution set of the given homogeneous system in parametric vector form.

6. \(\begin{array}{c}{x_1} + 3{x_2} - 5{x_3} = 0\\{x_1} + 4{x_2} - 8{x_3} = 0\\ - 3{x_1} - 7{x_2} + 9{x_3} = 0\end{array}\)

Short Answer

Expert verified

The general solution in the parametric vector form is represented as \(x = {x_3}\left[ {\begin{array}{*{20}{c}}{ - 4}\\3\\1\end{array}} \right].\)

Step by step solution

01

Convert the given system of equations into an augmented matrix

Anaugmented matrix for a system of equations is a matrix of numbers in which eachrow represents theconstants from one equation, and eachcolumn represents all thecoefficients for a single variable.

The augmented matrix \(\left[ {\begin{array}{*{20}{c}}A&0\end{array}} \right]\) for the given system of equations \({x_1} + 3{x_2} - 5{x_3} = 0,{x_1} + 4{x_2} - 8{x_3} = 0\) and \( - 3{x_1} - 7{x_2} + 9{x_3} = 0\) is represented as:

\(\left[ {\begin{array}{*{20}{c}}1&3&{ - 5}&0\\1&4&{ - 8}&0\\{ - 3}&{ - 7}&9&0\end{array}} \right]\)

02

Apply row operation

Perform an elementary row operation to produce the first augmented matrix.

Perform the sum of \( - 1\) times row 1 and row 2 at row 2 and the sum of \(3\) times row 1 and row 3 at row 3.

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

03

Apply row operation

Perform an elementary row operation to produce the second augmented matrix.

Perform the sum of \( - 2\) times row 2 and row 3 at row 3.

\(\left[ {\begin{array}{*{20}{c}}1&3&{ - 5}&0\\0&1&{ - 3}&0\\0&0&0&0\end{array}} \right]\)

04

Apply row operation

Perform an elementary row operation to produce the third augmented matrix.

Perform the sum of \( - 3\) times row 2 and row 1 at row 1.

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

05

Convert the matrix into the equation

To obtain the solution of the system of equations, you have to convert the augmented matrix into the system of equations again.

Write the obtained matrix \(\left[ {\begin{array}{*{20}{c}}1&0&4&0\\0&1&{ - 3}&0\\0&0&0&0\end{array}} \right]\)into the equation notation.

\[\begin{array}{c}{x_1} + 4{x_3} = 0\\{x_2} - 3{x_3} = 0\end{array}\]

Thus, \({x_1} = - 4{x_3},{x_2} = 3{x_3}\) and \({x_3}\) is a free variable.

06

Determine the general solution in the parametric vector form

The general solution of\(Ax = 0\) in the parametric vector form can be represented as:

\(\begin{array}{c}x = \left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}{ - 4{x_3}}\\{3{x_3}}\\{{x_3}}\end{array}} \right]\\ = {x_3}\left[ {\begin{array}{*{20}{c}}{ - 4}\\3\\1\end{array}} \right]\end{array}\)

Thus, the general solution in the parametric vector form is \(x = {x_3}\left[ {\begin{array}{*{20}{c}}{ - 4}\\3\\1\end{array}} \right]\).

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Most popular questions from this chapter

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?

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

12. \(\left[ {\begin{array}{*{20}{c}}1&{ - 7}&0&6&5\\0&0&1&{ - 2}&{ - 3}\\{ - 1}&7&{ - 4}&2&7\end{array}} \right]\).


Consider two vectors v鈫�1 andv鈫�2in R3 that are not parallel.

Which vectors inlocalid="1668167992227" 鈩�3are linear combinations ofv鈫�1andv鈫�2? Describe the set of these vectors geometrically. Include a sketch in your answer.

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

Consider the problem of determining whether the following system of equations is consistent for all \({b_1},{b_2},{b_3}\):

\(\begin{aligned}{c}{\bf{2}}{x_1} - {\bf{4}}{x_2} - {\bf{2}}{x_3} = {b_1}\\ - {\bf{5}}{x_1} + {x_2} + {x_3} = {b_2}\\{\bf{7}}{x_1} - {\bf{5}}{x_2} - {\bf{3}}{x_3} = {b_3}\end{aligned}\)

  1. Define appropriate vectors, and restate the problem in terms of Span \(\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3}} \right\}\). 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.
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