# A5S - ASSIGNMENT 5 Solutions 1[3 marks Prove that V = r ∈...

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Unformatted text preview: ASSIGNMENT 5 Solutions 1. [3 marks] Prove that V = { r ∈ R : r > } is a vector space under the following operations: r 1 + r 2 = r 2 r 1 (“addition”) and a · r = r a (“scalar multiplication”) . 1. (Here we have r 1 , r 2 , r ∈ V and a ∈ R .) Let a, b ∈ R and r 1 , r 2 , r 3 ∈ V . We show that the ten axioms of vector spaces hold.- r 1 + r 2 = r 2 r 1 ∈ V , since the product of two positive real numbers is a positive real number.- r 1 + r 2 = r 2 r 1 = r 1 r 2 = r 2 + r 1 .- ( r 1 + r 2 ) + r 3 = ( r 2 r 1 ) + r 3 = r 3 r 2 r 1 = ( r 3 r 2 ) r 1 = r 1 + ( r 3 r 2 ) = r 1 + ( r 2 + r 3 ) .- There exists an additive identity 1 ∈ V such that 1 + r 1 = r 1 (1) = r 1 .- Each element has an additive inverse:- r 1 = r- 1 1 ∈ V , since the reciprocal of a positive real number is a positive real number.- a · r 1 = r a 1 ∈ V , since the real power of a positive real number is a positive real number....
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A5S - ASSIGNMENT 5 Solutions 1[3 marks Prove that V = r ∈...

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