Good evening everyone, I hope everyone is having a better evening than myself thanks to this homework problem. Let P be the set of positive numbers. For a,b in P, define a+b=a x b; for a in P and a real number r, define r x a= a^r. Show that P is a vector space using ⊕ as addition and (circle dot) as scalar multiplication. For a,b in P, define a⊕b=a x b; for a in P and a real number r, define r x a= a^r. My professor wants us to use these properties that are in our text, Properties: 1. X+Y=Y+X. 2. (X+Y)+Z=X+(Y+Z). 3. 0+X=X+0=X. 4. r(sX)=(rs)X. 5. (r+s)X=rX+sX. 6. r(X+Y)=rX+rY. 7. 1X=X. My professor introduced our class to the topic of a vector space today and when he was talking about it everything made sense. Now that I am here on my own I honestly do not know where to start. Unfortunately I was unable to go to his office hours today to ask him about it. A brief and general description of where I should start is all I ask.