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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

Suppose \(a,b,c,\) and \(d\) are constants such that \(a\) is not zero and the system below is consistent for all possible values of \(f\) and \(g\). What can you say about the numbers \(a,b,c,\) and \(d\)? Justify your answer.

28. \(\begin{array}{l}a{x_1} + b{x_2} = f\\c{x_1} + d{x_2} = g\end{array}\)

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  1. \(\begin{aligned}{l}2{x_1} - 5{x_2} + 8{x_3} = 0\\ - 2{x_1} - 7{x_2} + {x_3} = 0\\4{x_1} + 2{x_2} + 7{x_3} = 0\end{aligned}\)

In Exercises 5, write a system of equations that is equivalent to the given vector equation.

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In Exercise 22, mark each statement True or False. Justify each answer.

22. a. Every matrix transformation is a linear transformation.

b. The codomain of the transformation \({\bf{x}} \mapsto {\bf{Ax}}\) is the set of all linear combinations of the columns of \({\bf{A}}\).

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