Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 1: Q20E (page 1)

Determine by inspection whether the vectors in Exercises 15-20 are linearly independent. Justify each answer.

20. \(\left[ {\begin{array}{*{20}{c}}1\\4\\{ - 7}\end{array}} \right],\left[ {\begin{array}{*{20}{c}}{ - 2}\\5\\3\end{array}} \right],\left[ {\begin{array}{*{20}{c}}0\\0\\0\end{array}} \right]\)

Short Answer

Expert verified

The set is linearly dependent.

Step by step solution

01

Determine whether the vectors are multiples of each other

No vector in the set is a multiple of other vectors.

02

Determine whether the set contains more vectors than the entries

Theorem 8 tells that if a set contains more vectors than entries in each vector, then the set is linearlydependent.

The theorem does not apply here because the number of vectors does not exceed the number of entries in each vector.

03

Determine whether the vectors are linearly independent by inspection

Theorem 9tells that if a set \(S = \left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},...,{{\mathop{\rm v}\nolimits} _p}} \right\}\) in \({\mathbb{R}^n}\) contains the zero vector, then it is linearly dependent.

Here, the set contains the zero vector. Therefore, it is linearly dependent.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

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

Determine the values(s) of \(h\) such that matrix is the augmented matrix of a consistent linear system.

17. \(\left[ {\begin{array}{*{20}{c}}2&3&h\\4&6&7\end{array}} \right]\)

Use Theorem 7 in section 1.7 to explain why the columns of the matrix Aare linearly independent.

\(A = \left( {\begin{aligned}{*{20}{c}}1&0&0&0\\2&5&0&0\\3&6&8&0\\4&7&9&{10}\end{aligned}} \right)\)

Solve the linear system of equations. You may use technology.

|3x+5y+3z=257X+9y+19z=654X+5y+11z=5|

In Exercises 11 and 12, determine if \({\rm{b}}\) is a linear combination of \({{\mathop{\rm a}\nolimits} _1},{a_2}\) and \({a_3}\).

12.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free