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    I have this question, which I'm really stuck on...

     \mathrm{ Show\ if\ the\ set\ Q\ of\ pairs\ of\ positive\ real\ numbers}

    Q = \{(x,y)\lvert x,y > 0\} \subset \mathbb{R}^2

     \mathrm{ is\ a\ real\ vector\ space\ if\ we\ define\ addition\ by\ (x,y)\ +\ (x',y')\ =\ (x \times x' , y \times y')}

     \mathrm{ and\ scalar\ multiplication\ by\ \lambda (x,y)\ = (x^\lambda, y^\lambda)}

    Now, is this a case of using the eight axioms to show this?

    I know this sounds a stupid question, as I'm almost sure it is, but my lecturer is one for throwing us something that looks right, up until the end when you realise you've been chasing a dead end :grr:

    I should add, the reason I'm not sure, is I cannot find anything in my notes about defining addition or scalar multiplication in these manners, and so wouldn't know where to start...

    Any help would be appreciated
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    (Original post by ekudamram)
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    Yes, all you need to do is run through the axioms.

    Here's commutivity of addition as an example.


    (x,y)+(x',y')=(x\times x',y\times y') by definition of vector addition in Q

    =(x'\times x,y'\times y) Since multiplication in \mathbb{R} is commutative.

    =(x',y')+(x,y) and again by definition of vector addition in Q.

    And now you've shown commutivity in your vector space.


    Edit: Seems that what your lecturer is trying to do here is to get you to appreciate the difference between the vector space axioms, and how vectors are represented.
 
 
 
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