hw4 - Solutions of Selected Problems in HW4 3.1.12 Let x, y...

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Solutions of Selected Problems in HW4 3.1.12 Let x,y be arbitrary elements of R + . Also let α,β be arbitrary real numbers. C1: x y = x · y R + , since multiplication of two positive numbers are also positive. C2: α x = x α R + , since exponential functions with positive base are always positive. A1: x y = x · y = y · x = y x. A2: ( x y ) z = ( x · y ) · z = x · ( y · z ) = x ( y z ) . A3: In this space, 1 plays a role of a zero vector with respect to , since 1 x = 1 · x = x. A4: In this space, 1 x plays a role of an inverse of x with respect to , since x 1 x = x · 1 x = 1 . Note that 1 is the zero vector. A5: α ( x y ) = α ( x · y ) = ( x · y ) α = x α · y α = ( α x ) · ( α y ) = ( α x ) ( α y ) A6: ( α + β ) x = x α + β = x α · x β = ( α x ) · ( β x ) = ( α x ) ( β x ) . A7: (
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hw4 - Solutions of Selected Problems in HW4 3.1.12 Let x, y...

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