Explanation Given: The set V and S are as follows, V = ℝ 4 , S = { ( 6 , − 3 , 2 , 0 ) , ( 1 , 1 , 1 , 0 ) , ( 1 , − 8 , − 1 , 0 ) } . Approach: A set of vectors { v 1 , v 2 , … , v k } in a vector space V spans V if every vector in V can be written as a linear combination of v 1 , v 2 , … , v k ; that is, if for every v ∈ V , there exists scalars c 1 , c 2 , … , c k such that v = c v 1 + c v 2 + … + c v k . Calculation: Consider an arbitrary vector ( x , y , z , w ) ∈ ℝ 4 = V Then, ( x , y , z , w ) = a ( 6 , − 3 , 2 , 0 ) + b ( 1 , 1 , 1 , 0 ) + c ( 1 , − 8 , − 1 , 0 ) = ( 6 a , − 3 a , 2 a , 0 a ) + ( 1 b , 1 b , 1 b , 0 b ) + ( 1 c , − 8 c , − 1 c , 0 c ) = ( 6 a + b + c , − 3 a + b − 8 c , 2 a + b − c , 0 ) for a , b , c ∈ ℝ Compare both sides. This gives, 6 a + b + c = x − 3 a + b − 8 c = y 2 a + b − c = z 0 a + 0 b + 0 c = w The augmented matrix of the above system is, [ 6 1 1 − 3 1 − 8 2 1 − 1 0 0 0 | x y z w ] Apply the row operation R 2 → R 2 + R 3 . [ 6 1 1 − 1 2 − 9 2 1 − 1 0 0 0 | x y + z z w ] Apply the row operation R 1 ↔ R 2 To determine (b) To find Whether the set S = { ( 6 , − 3 , 2 , 0 ) , ( 1 , 1 , 1 , 0 ) , ( 1 , − 8 , − 1 , 0 ) } of vectors is linearly independent or not.

Differential Equations and Linear Algebra (4th Edition)

4th Edition

ISBN: 9780321964670

Author: Stephen W. Goode, Scott A. Annin

Publisher: PEARSON

See similar textbooks

Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied unde…

Videos

Question

Chapter 4.11, Problem 26AP

To determine

(a)

To find:

Whether S={(6,−3,2,0),(1,1,1,0),(1,−8,−1,0)} spans V=ℝ4.

To determine

(b)

To find

Whether the set S={(6,−3,2,0),(1,1,1,0),(1,−8,−1,0)} of vectors is linearly independent or not.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4.11, Problem 25AP

Chapter 4.11, Problem 27AP

Students have asked these similar questions

Which graph represents f(x) = √x-2+3?

Practice Assignment 5.6 Rational Functions M Practice Assig Practice Assignment 5.6 Rational Functions Score: 120/150 Answered: 12/15 Question 10 A Write an equation for the function graphed below 5 + 4 1 2 H + + -7 -6 -5 -4 -3 -2 -1 2 34567 | -2 ర y = Question Help: Video Message instructor Post to forum Submit Question

it's not algebra 4th grade

Chapter 4 Solutions

Differential Equations and Linear Algebra (4th Edition)

Ch. 4.1 - True-False Review For items a-I, decide if the...Ch. 4.1 - Prob. 2TFR Ch. 4.1 - For items a-I, decide if the given statement is...Ch. 4.1 - For items a-I, decide if the given statement is...Ch. 4.1 - For items a-I, decide if the given statement is...Ch. 4.1 - For items a-I, decide if the given statement is...Ch. 4.1 - Prob. 7TFR Ch. 4.1 - Prob. 8TFR Ch. 4.1 - For items a-I, decide if the given statement is...Ch. 4.1 - Prob. 10TFR

Knowledge Booster

$Algebra$

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.

(a) To find: Whether S = { ( 6 , − 3 , 2 , 0 ) , ( 1 , 1 , 1 , 0 ) , ( 1 , − 8 , − 1 , 0 ) } spans V = ℝ 4 .

Solution Summary: The author explains that the set of vectors does not span V if every vector can be written as a linear combination of v_1,

Concept explainers

Videos

(a)

To find:

Whether S={(6,−3,2,0),(1,1,1,0),(1,−8,−1,0)} spans V=ℝ4.

(b)

To find

Whether the set S={(6,−3,2,0),(1,1,1,0),(1,−8,−1,0)} of vectors is linearly independent or not.

Want to see the full answer?

Chapter 4 Solutions

(a) To find: Whether S = { ( 6 , − 3 , 2 , 0 ) , ( 1 , 1 , 1 , 0 ) , ( 1 , − 8 , − 1 , 0 ) } spans V = ℝ 4 .

Solution Summary: The author explains that the set of vectors does not span V if every vector can be written as a linear combination of v_1,

Concept explainers

Videos

(a) To find: Whether S={(6,−3,2,0),(1,1,1,0),(1,−8,−1,0)} spans V=ℝ4.

(b) To find Whether the set S={(6,−3,2,0),(1,1,1,0),(1,−8,−1,0)} of vectors is linearly independent or not.

Want to see the full answer?

Chapter 4 Solutions

(a)

To find:

Whether S={(6,−3,2,0),(1,1,1,0),(1,−8,−1,0)} spans V=ℝ4.

(b)

To find

Whether the set S={(6,−3,2,0),(1,1,1,0),(1,−8,−1,0)} of vectors is linearly independent or not.