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Chapter 1: Q36E (page 1)

Each statement in Exercises 33鈥�38 is either true (in all cases) or false (for at least one example). If false, construct a specific example to show that the statement is not always true. Such an example is called a counterexampleto the statement. If a statement is true, give a justification. (One specific example cannot explain why a statement is always true. You will have to do more work here than in Exercises 21 and 22.)

36. If \({{\bf{v}}_1},...,{{\bf{v}}_4}\) are in \({\mathbb{R}^4}\) and \({{\bf{v}}_3}\) is not a linear combination of \({{\bf{v}}_1}\), \({{\bf{v}}_2}\), \({{\bf{v}}_4}\), then \(\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3},{{\bf{v}}_4}} \right\}\) is linearly independent.

Short Answer

Expert verified

The statement is false.

Step by step solution

01

Write the condition for the set to be linearly dependent

The vectors are said to be linearly dependent if the equation \({c_1}{{\bf{v}}_1} + {c_2}{{\bf{v}}_2} + ... + {c_p}{{\bf{v}}_p} = 0\) has a non-trivial solution, where \({c_1},{c_2},...,{c_p}\) are scalars.

02

Write the cases for a linear combination

It is given that \({{\bf{v}}_3}\) is not a linear combination of \({{\bf{v}}_1}\), \({{\bf{v}}_2}\), and \({{\bf{v}}_4}\). Since \({{\bf{v}}_3}\) is not a linear combination, \({{\bf{v}}_3}\) does not have scalars. It is possible that u and v are linearly dependent in some circumstances. Then, it could be linearly dependent.

Thus, the statement is false.

03

Write the example according to the condition

Let the vectors be \({{\bf{v}}_1} = \left[ {\begin{array}{*{20}{c}}1\\1\\1\\1\end{array}} \right]\), \({{\bf{v}}_2} = \left[ {\begin{array}{*{20}{c}}2\\2\\2\\2\end{array}} \right]\), \({{\bf{v}}_3} = \left[ {\begin{array}{*{20}{c}}1\\0\\0\\0\end{array}} \right]\), and \({{\bf{v}}_4} = \left[ {\begin{array}{*{20}{c}}4\\4\\4\\4\end{array}} \right]\).

Here, it is observed that \({{\bf{v}}_4} = 2{{\bf{v}}_1} + {{\bf{v}}_2}\).

So, it shows that \({{\bf{v}}_3}\) is not a linear combination of \({{\bf{v}}_1}\), \({{\bf{v}}_2}\), and \({{\bf{v}}_4}\).

Hence, the statement is false.

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Most popular questions from this chapter

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.

Use the accompanying figure to write each vector listed in Exercises 7 and 8 as a linear combination of u and v. Is every vector in \({\mathbb{R}^2}\) a linear combination of u and v?

8.Vectors w, x, y, and z

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 \({{\mathop{\rm a}\nolimits} _2}\)?

Let \({{\mathop{\rm v}\nolimits} _1} = \left[ {\begin{array}{*{20}{c}}1\\0\\{ - 2}\end{array}} \right],{v_2} = \left[ {\begin{array}{*{20}{c}}{ - 3}\\1\\8\end{array}} \right],\) and \({\rm{y = }}\left[ {\begin{array}{*{20}{c}}h\\{ - 5}\\{ - 3}\end{array}} \right]\). For what values(s) of \(h\) is \(y\) in the plane generated by \({{\mathop{\rm v}\nolimits} _1}\) and \({{\mathop{\rm v}\nolimits} _2}\)

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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