Let F = R and W = {(#1, #2,... , tn) e R* | #1 + *2 +.+ & = 0}.(a) Let n = 4, so that W = {(*, y, z, w) E R* | * + y + z + w = 0}. Prove that {ei – e2, e2 - e3, eg - e4} is a basis of W.(b) Generalize this to general n: Prove that B = {ex – ex+1 | k = 1,2,... n – 1} is a basis for W.

Answered: Image /qna-images/answer/bf89cfb4-7076-449f-9ac1-7a326134ef13.jpg

Let IF = R and W = {(#1, #2,... , tn) € R* | #1 + *2 + · . · + &n = 0}. (a) Let n :4, so that W = {(*, 3, 2, w) e R* | # + y + z + w = 0}. Prove that {ei e2, e2 es, eg - e4} is a basis of W. (b) Generalize this to general n: Prove that B = {ex ex+1 | k = 1, 2, ... n – 1} is a basis for W.

Let IF = R and W = {(#1, #2,... , tn) € R* | #1 + 2 + · . · + &n = 0}. (a) Let n :4, so that W = {(, 3, 2, w) e R* | # + y + z + w = 0}. Prove that {ei e2, e2 es, eg - e4} is a basis of W. (b) Generalize this to general n: Prove that B = {ex ex+1 | k = 1, 2, ... n – 1} is a basis for W.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter8: Applications Of Trigonometry

Section8.3: Vectors

Problem 60E

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Question

Let F = R and W = {(#1, #2,... , tn) e R* | #1 + *2 +.+ & = 0}.
(a) Let n = 4, so that W = {(*, y, z, w) E R* | * + y + z + w = 0}. Prove that {ei – e2, e2 - e3, eg - e4} is a basis of W.
(b) Generalize this to general n: Prove that B = {ex – ex+1 | k = 1,2,... n – 1} is a basis for W.

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