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Chapter 1: Q33E (page 1)

Each statement in Exercises 33-38 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 counterexample to 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.)

33. If \({{\mathop{\rm v}\nolimits} _1},...,{v_4}\) are in \({\mathbb{R}^4}\) and \({{\mathop{\rm v}\nolimits} _3} = 2{{\mathop{\rm v}\nolimits} _1} + {v_2}\), then \(\left\{ {{v_1},{v_2},{v_3},{v_4}} \right\}\) is linearly dependent.

Short Answer

Expert verified

The given statement is true.

Step by step solution

01

Determine whether the given statement is true or false

An indexed set \(S = \left\{ {{{\mathop{\rm v}\nolimits} _1},...,{v_p}} \right\}\) of two or more vectors islinearly dependentif and only if one of the vectors in \(S\) is a linear combination of the others.

Thus, the given statement is true.

02

Explain why the given statement is true

The given linear combination of vectors in \({\mathbb{R}^4}\) is \({{\mathop{\rm v}\nolimits} _3} = 2{{\mathop{\rm v}\nolimits} _1} + {v_2}\).

Since one of the vectors in \({\mathbb{R}^4}\) is a linear combination of the others, the vectors in \({\mathbb{R}^4}\) are linearly dependent.

Thus, the given statement is true.

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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.

Consider a dynamical system x→(t+1)=Ax→(t) with two components. The accompanying sketch shows the initial state vector x→0and two eigen vectors υ1→  and  υ2→ of A (with eigen values λ1→andλ2→ respectively). For the given values of λ1→andλ2→, draw a rough trajectory. Consider the future and the past of the system.

λ1→=1,λ2→=0.9

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 thatis the vertex of a parallelogram whose other vertices are u, 0, and \( - {\bf{v}}\).

3. u and v as in Exercise 1

Write the vector \(\left( {\begin{array}{*{20}{c}}5\\6\end{array}} \right)\) as the sum of two vectors, one on the line \(\left\{ {\left( {x,y} \right):y = {\bf{2}}x} \right\}\) and one on the line \(\left\{ {\left( {x,y} \right):y = x/{\bf{2}}} \right\}\).

Construct three different augmented matrices for linear systems whose solution set is \({x_1} = - 2,{x_2} = 1,{x_3} = 0\).
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