Question

How do we prove that this is a vector space?
The set of real numbers R+
Addition: x+y = xy
Scalar: rx = x^r

Answer 1

hey so you wanna show that the set of real numbers is a vector space? or real numbers applied to these rules is a vector spae?

Answer 2

a vector space is when, any 2 given 'vectors' under your scalar rule will produce another 'vector' in your 'vector' space

Answer 3

@518nad yeah i wanna show that the set of real numbers is a vector space. I'm a little confused by the concept to say "yes these set of numbers is a vector space". It's a little tricky for me to understand

Answer 4

okay so lets see if your set of real numbers first satisfy your scalar rule

Answer 5

we have to show that any possible real number we produce with the scalar rule will definitely satisfy this scalar rule

Answer 6

start with 1=r, x=2
1*2=2^1 true
now
2*2=2^2
4=4
okay since we saw that 4 is in this vector space, that means
there must exist another number like
r*4=4^r
in this case r=1 is the only solution

Answer 7

actually i dont know if that really is the only solution. there might be more let me see

Answer 8

4^r-4r=0
there are infinite solutions to this equations i believe, and you need to show that every one of these solutions exist

Answer 9

okay lets start with the general case then

Answer 10

oh no only 2 solutions, since this has to be a real solutions only

Answer 11

hi

Answer 12

okay @518nad i remember my professor mentioning 1 being one of the solutions. It's nice to see it this way. He never rly substituted actual numbers to emphasize the equation.

Answer 13

whats the question

Answer 14

it's at the top @HelpWithK12

Answer 15

ok

Answer 16

pretty cool question

Answer 17

yeah quit itrsting but i can do it

Answer 18

hmmmmm

Answer 19

x+y=xy