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,...
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![If the vector addition of a vector space V has been defined to be: ū©v = (µ₁ +v₁, U₂+v₂, Uz+vž) \ū‚Ñ€V,
and the scalar multiplication of V has been defined to be: a v = (v₁₂ √₂, α²) \а€R‚¯Ñ€V,
then which basic properties in above problem are not violated?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6744edbe-de6d-4c3b-b37d-23bf1fe16ab7%2F7570926b-eb5c-4cfc-805c-956f12f93a3d%2F9ahmgvl_processed.png&w=3840&q=75)
Transcribed Image Text:If the vector addition of a vector space V has been defined to be: ū©v = (µ₁ +v₁, U₂+v₂, Uz+vž) \ū‚Ñ€V,
and the scalar multiplication of V has been defined to be: a v = (v₁₂ √₂, α²) \а€R‚¯Ñ€V,
then which basic properties in above problem are not violated?
![10 basic properties of vector space
Vā,b,c eV,Vc,kɛR.
(1) Closedness of addition: Vā,b €V ⇒ā + b €V.
(2) Commutativity of addition: a+b=b+ā
(3) Associativity of addition: (a+b)+c = à +(b + c)
(4) addition of zero vector: à +0= ā
(5) Negativity of addition: ā +(-a)=0
(6) Closedness of scalar multiplication: Va eV, Vce R⇒cā EV.
(7) Distributivity of scalar multiplication versus addition:c(a+b)=cā + cb
(8) Distributivity of addition versus scalar multiplication: (c+k)ā = cā + kā
(9) Associativity of scalar multiplication: c(kā) = (ck)ā
(10) Unit element of scalar multiplication: là = ā
13](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6744edbe-de6d-4c3b-b37d-23bf1fe16ab7%2F7570926b-eb5c-4cfc-805c-956f12f93a3d%2Fsqxud6a_processed.png&w=3840&q=75)
Transcribed Image Text:10 basic properties of vector space
Vā,b,c eV,Vc,kɛR.
(1) Closedness of addition: Vā,b €V ⇒ā + b €V.
(2) Commutativity of addition: a+b=b+ā
(3) Associativity of addition: (a+b)+c = à +(b + c)
(4) addition of zero vector: à +0= ā
(5) Negativity of addition: ā +(-a)=0
(6) Closedness of scalar multiplication: Va eV, Vce R⇒cā EV.
(7) Distributivity of scalar multiplication versus addition:c(a+b)=cā + cb
(8) Distributivity of addition versus scalar multiplication: (c+k)ā = cā + kā
(9) Associativity of scalar multiplication: c(kā) = (ck)ā
(10) Unit element of scalar multiplication: là = ā
13
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