Show-by-use-of-the-characteristics-of-a-vector-space-that-the-set-x-R-with-the-following-operations-and-builds-a-vector-space-Use-vector-space-axioms-x-y-be-x-y-for-R

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Question Number 1537 by 2closedStringsMeet last updated on 17/Aug/15

Show by use of the characteristics of a vector−space, that the set x^→ ∈R_+ with the following operations ⊕ and builds a vector−space. ∗Use vector−space axioms ∗x^→ ⊕y^→ be x^→ ∙y^→ ∗for λ∈R set λ x^→ equal x^(→λ)

$${Show}\:{by}\:{use}\:{of}\:{the}\:{characteristics}\:{of} \\ $$$${a}\:{vector}−{space},\:{that}\:{the}\:{set}\:\overset{\rightarrow} {{x}}\in\mathbb{R}_{+} \\ $$$$\mathrm{with}\:\mathrm{the}\:\mathrm{following}\:\mathrm{operations}\:\oplus\:{and}\: \\ $$$${builds}\:{a}\:{vector}−{space}. \\ $$$$\ast{Use}\:{vector}−{space}\:{axioms} \\ $$$$\ast\overset{\rightarrow} {{x}}\oplus\overset{\rightarrow} {{y}}\:\:\:\:{be}\:\:\:\overset{\rightarrow} {{x}}\centerdot\overset{\rightarrow} {{y}} \\ $$$$\ast{for}\:\lambda\in\mathbb{R}\:\:{set}\:\:\lambda \overset{\rightarrow} {{x}}\:\:{equal}\:\overset{\rightarrow\lambda} {{x}} \\ $$

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Show-by-use-of-the-characteristics-of-a-vector-space-that-the-set-x-R-with-the-following-operations-and-builds-a-vector-space-Use-vector-space-axioms-x-y-be-x-y-for-R

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