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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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