Determine if V is a vector space when:
x+y=xy
cx=x^c
If not state all the vector axioms it fails

Question

Determine if V is a vector space when:
x+y=xy
cx=x^c
If not state all the vector axioms it fails

anonymous · Answer

If yes verify each vector space axiom

turingtest · Answer

here's at least one that fails$$c(\vec u+\vec v)\neq c\vec u+c\vec v$$

anonymous · Answer

ok what abt tat scaler multiplication?

phi · Answer

x^c is not a linear operation

anonymous · Answer

okkkkk

turingtest · Answer

actually it's consistent though...
$$c(\vec u+\vec v)= c\vec u+c\vec v$$because we have$$c(\vec u+\vec v)=(uv)^c=u^cv^c= c\vec u+c\vec v$$so I don't see what's wrong with it

anonymous · Answer

well it is definitely not closed under scaler multilication though

turingtest · Answer

how so?

anonymous · Answer

because x^c is exponential

turingtest · Answer

but that doesn't necessarily take it out of V

turingtest · Answer

*the vector space I mean

anonymous · Answer

oh ya? ok I dont get a thing, I feel so stupid

turingtest · Answer

aha! it is a vector space I thought I'd seen it befor check it out, the zero vector turns out to be 1 I think http://tutorial.math.lamar.edu/Classes/LinAlg/VectorSpaces.aspx

anonymous · Answer

yay lol :D

phi · Answer

@Tur how do you get from (uv)^c to cu + cv?

turingtest · Answer

check out example 5

anonymous · Answer

yup i see it. Thanks :D

turingtest · Answer

@ phi that is what you get when you put the two rules together
the exponent is distributed because vector addition is defined as multiplication

phi · Answer

nvm, i'll look at your link...

anonymous · Answer

LOL i htink i am fundamentally lacking basic knowledge

turingtest · Answer

I'm not so good at this stuff, I had it wrong at first as well