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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

Let \(T:{\mathbb{R}^2} \to {\mathbb{R}^2}\) be the linear transformation such that \(T\left( {{e_1}} \right)\) and \(T\left( {{e_2}} \right)\) are the vectors shown in the figure. Using the figure, sketch the vector \(T\left( {2,1} \right)\).

a. Find the general flow pattern in the network shown in the figure.

b. Assuming that the flow must be in the directions indicated, find the minimum flows in the branches denoted by \({x_2}\), \({x_3}\), \({x_4}\) and \({x_5}\).

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.

Give a geometric description of Span \(\left\{ {{v_1},{v_2}} \right\}\) for the vectors in Exercise 16.

In a certain region, about 6% of city鈥檚 population moves to the surrounding suburbs each year, and about 4% of the suburban population moves into the city. In 2015, there were, 10,000,000 residents in the city and 800,000 in the suburbs. Set up a difference equation that describes this situation, where\({{\bf{x}}_{\bf{0}}}\)is the initial population in 2015. Then estimate the population in the city and in the suburbs two years later, in 2017.
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