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Chapter 1: Q21E (page 1)

In Exercises 21 and 22, mark each statement True or False. Justify each answer on the basis of a careful reading of the text.

21.

a. The columns of a matrix \(A\) are linearly independent if the equation \(Ax = 0\) has a trivial solution.

b. If \(S\) is a linearly dependent set, then each vector is a linear combination of the other vectors in \(S\).

c. The columns of any \(4 \times 5\) matrix are linearly dependent.

d. If \({\mathop{\rm x}\nolimits} \) and \(y\) are linearly independent, and if \(\left\{ {x,y,z} \right\}\) is linearly dependent, then \(z\) is in Span\(\left\{ {x,y} \right\}\).

Short Answer

Expert verified
  1. The given statement is false.
  2. The given statement is false.
  3. The given statement is true.
  4. The given statement is true.

Step by step solution

01

Identify whether the given statement is true or false

a.

The columns of matrix \(A\) are linearly independent if and only if the equation \(Ax = 0\) has only a trivial solution.

Thus, statement (a) is false.

02

Identify whether the given statement is true or false

b.

A vector in a linearly dependent set may fail to be a linear combination of other vectors.

Thus, statement (b) is false.

03

Identify whether the given statement is true or false

c.

Let \(A\) be a \(n \times p\) matrix. If \(p > n\), the columns are linearly dependent.

Thus, statement (c) is true.

04

Identify whether the given statement is true or false

d.

Any set \(\left\{ {u,v,w} \right\}\) in \({\mathbb{R}^3}\) with, linearly independent vectors \({\mathop{\rm u}\nolimits} \) and \({\mathop{\rm v}\nolimits} \), is linearly dependent if and only if w is in the plane spanned by \({\mathop{\rm u}\nolimits} \) and \({\mathop{\rm v}\nolimits} \).

Thus, statement (d) is true.

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Most popular questions from this chapter

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 \({{\mathop{\rm v}\nolimits} _2}\)

Let \(T:{\mathbb{R}^n} \to {\mathbb{R}^n}\) be an invertible linear transformation. Explain why T is both one-to-one and onto \({\mathbb{R}^n}\). Use equations (1) and (2). Then give a second explanation using one or more theorems.

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

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.

Determine whether the statements that follow are true or false, and justify your answer.

18: [111315171921][-13-1]=[131921]
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