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Chapter 1: Q24E (page 1)

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

24.

a. If \(x\) is a nontrivial solution of \(Ax = 0\), then every entry in \({\mathop{\rm x}\nolimits} \) is nonzero.

b. The equation \(x = {x_2}{\mathop{\rm u}\nolimits} + {x_3}{\mathop{\rm v}\nolimits} \) , with \({x_2}\) and \({x_3}\) free (and neither \({\mathop{\rm u}\nolimits} \) nor \(v\) a multiple of the other), describes a plane through the origin.

c. The equation \(Ax = b\) is homogeneous if the zero vector is a solution.

d. The effect of adding \({\mathop{\rm p}\nolimits} \) to a vector is to move the vector in a direction parallel to \({\mathop{\rm p}\nolimits} \).

e. The solution set of \(Ax = b\) is obtained by translating the solution set of \(Ax = 0\).

Short Answer

Expert verified

a. The given statement isfalse.

b. The given statement istrue.

c. The given statement istrue.

d. The given statement istrue.

e. The given statement isfalse.

Step by step solution

01

(a) Identify whether the statement is true or false

a.

Anontrivial solution \(x\) can have some zero entries until and unless all of its entries are zero.

Thus, the given statement (a) is false.

02

(b) Identify whether the statement is true or false

b.

As it is known that neither \({\mathop{\rm u}\nolimits} \) nor \({\mathop{\rm v}\nolimits} \) is a scalar multiple of the other, a solution set is a plane through the origin.

Thus, the given statement (b) is true.

03

(c) Identify whether the statement is true or false

c.

A homogeneous linear system \(Ax = 0\) always has at least one solution \(x = 0\). Zero vector is a solution because \(b = Ax = A0 = 0\) .

Thus, the given statement (c) is true.

04

(d) Identify whether the statement is true or false

d.

It is known that the effect of adding \({\mathop{\rm p}\nolimits} \) to \({\mathop{\rm v}\nolimits} \) moves \({\mathop{\rm v}\nolimits} \) in a direction parallel to the line through 0 and \({\mathop{\rm v}\nolimits} \).

Thus, the given statement (d) is true.

05

(e) Identify whether the statement is true or false

e.

The solution set is obtained by translating the solution set of \(Ax = 0\). The statement holds only if the solution set \({\mathop{\rm Ax}\nolimits} = 0\) is nonempty.

Thus, the given statement (e) is false.

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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 de铿乶ition 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.
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