/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 49 By inspection, determine why eac... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 49

By inspection, determine why each of the sets is linearly dependent. (a) \(S=\\{(1,-2),(2,3),(-2,4)\\}\) (b) \(S=\\{(1,-6,2),(2,-12,4)\\}\) (c) \(S=\\{(0,0),(1,0)\\}\)

Short Answer

Expert verified
(a) The set \(S=\{(1,-2),(2,3),(-2,4)\}\) is linearly independent as no vector can be expressed as a linear combination of the others. (b) The set \(S=\{(1,-6,2),(2,-12,4)\}\) is linearly dependent because one vector can be expressed twice the other. (c) The set \(S=\{(0,0),(1,0)\}\) is linearly dependent since it includes the zero vector.

Step by step solution

01

Part (a): S={1,-2),(2,3),(-2,4)}

In this set, one can notice that the third vector can be obtained by subtracting second vector from the first one i.e., (1, -2) - (2, 3) = (-1, -5), not equal to (-2, 4). Thus, none of the vectors are a linear combination of other vectors. Hence, any previous belief about this set being linearly dependent was incorrect. In reality, it is linearly independent.
02

Part (b): S={(1,-6,2),(2,-12,4)}

For these vectors, it can be seen that the second vector is twice the first vector i.e., 2*(1,-6,2) = (2,-12,4). Therefore, the set is linearly dependent because one vector can be represented as a linear combination of other vectors. The second vector does not provide any additional information beyond that provided by the first vector.
03

Part (c): S={(0,0),(1,0)}

In this set, the first vector is a zero vector, and note that a set with a zero vector is always linearly dependent. The zero vector can be expressed as a linear combination of any vector because 0 times any vector equals the zero vector.

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

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

Use a graphing utility or computer software program with matrix capabilities to find the transition matrix from \(B\) to \(B^{\prime}\) $$B=\\{(2,5),(1,2)\\}, B^{\prime}=\\{(2,1),(-1,2)\\}$$

Find the coordinate matrix of \(X\) relative to the standard basis in \(M_{3,1}\) $$X=\left[\begin{array}{r}2 \\ -1 \\ 4\end{array}\right]$$

Test the given set of solutions for linear independence. $$\begin{array}{lll} \text { Differential Equation } & \text { Solutions } \\ y^{\prime \prime \prime \prime}+3 y^{\prime \prime}+3 y^{\prime}+y=0 & \left\\{e^{-x}, x e^{-x}, x^{2} e^{-x}\right\\} \end{array}$$

Identify and sketch the graph. $$\frac{x^{2}}{16}-\frac{y^{2}}{25}=1$$

Perform a rotation of axes to eliminate the \(x y\) -term, and sketch the graph of the conic. $$13 x^{2}+6 \sqrt{3} x y+7 y^{2}-16=0$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Decision Maths

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.