Homework Help is Here – Start Your Trial Now!
Math
Algebra
Let S = {(1,0, 1), (0, 1,0)} be two vectors in R³. Write the following vectors as a linear combination of the vectors in S if possible. If it's not possible, show the matrix indicating why it isn't. (a) (-4, 6, –4) (b) (-4,3, 3) (c) (0,24, 0) (d) (1,0,0)

Let S = {(1,0, 1), (0, 1,0)} be two vectors in R³. Write the following vectors as a linear combination of the vectors in S if possible. If it's not possible, show the matrix indicating why it isn't. (a) (-4, 6, –4) (b) (-4,3, 3) (c) (0,24, 0) (d) (1,0,0)

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
See similar textbooks
icon
Related questions
Question
**Vector Combinations in \( \mathbb{R}^3 \)** Let's consider the set \( S = \{(1, 0, 1), (0, 1, 0)\} \) consisting of two vectors in \( \mathbb{R}^3 \). Our goal is to express the following vectors as linear combinations of the vectors in \( S \). If it is not feasible to do so, we will show the matrix that explains why it is not possible. (a) \((-4, 6, -4)\) (b) \((-4, 3, 3)\) (c) \((0, 24, 0)\) (d) \((1, 0, 0)\) **Explanation:** We aim to solve for constants \( a \) and \( b \) such that each of the given vectors can be written in the form: \[ a(1, 0, 1) + b(0, 1, 0) = (a, b, a) \] For each vector: 1. Determine \( a \) and \( b \) if possible. 2. Present the matrix representation that confirms whether the solutions exist. This structured approach helps in verifying and understanding the linear dependence of vectors in \( \mathbb{R}^3 \).
Transcribed Image Text:**Vector Combinations in \( \mathbb{R}^3 \)** Let's consider the set \( S = \{(1, 0, 1), (0, 1, 0)\} \) consisting of two vectors in \( \mathbb{R}^3 \). Our goal is to express the following vectors as linear combinations of the vectors in \( S \). If it is not feasible to do so, we will show the matrix that explains why it is not possible. (a) \((-4, 6, -4)\) (b) \((-4, 3, 3)\) (c) \((0, 24, 0)\) (d) \((1, 0, 0)\) **Explanation:** We aim to solve for constants \( a \) and \( b \) such that each of the given vectors can be written in the form: \[ a(1, 0, 1) + b(0, 1, 0) = (a, b, a) \] For each vector: 1. Determine \( a \) and \( b \) if possible. 2. Present the matrix representation that confirms whether the solutions exist. This structured approach helps in verifying and understanding the linear dependence of vectors in \( \mathbb{R}^3 \).
Expert Solution
steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Knowledge Booster
Vector Space
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra And Trigonometry (11th Edition)
Algebra And Trigonometry (11th Edition)
Algebra
ISBN:
9780135163078
Author:
Michael Sullivan
Publisher:
PEARSON
Introduction to Linear Algebra, Fifth Edition
Introduction to Linear Algebra, Fifth Edition
Algebra
ISBN:
9780980232776
Author:
Gilbert Strang
Publisher:
Wellesley-Cambridge Press
College Algebra (Collegiate Math)
College Algebra (Collegiate Math)
Algebra
ISBN:
9780077836344
Author:
Julie Miller, Donna Gerken
Publisher:
McGraw-Hill Education
SEE MORE TEXTBOOKS