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

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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…

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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.

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Chapter 4.11, Problem 25AP

Chapter 4.11, Problem 27AP

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Chapter 4 Solutions

Differential Equations and Linear Algebra (4th Edition)

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(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,

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(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.