Question

Let V be the set of all positive real numbers with vector addition being ordinary multiplication, and scalar multiplication being a.v=v^a. Show that V is a vector space. Thanks in advance.

Answer 1

@ikram002p hihihi shame on me, I know it is vector space but forgot how to prove.

Answer 2

a(x+y) = ax+ay = x^a + y^a

Answer 3

v1 and v2 belong to v
_ v1 + v2 = v1.v2 (specific rule) positive real => true
_ a.v1=v1^a positive real => true
_ v1 belong to V, -v1=-1.v1=v1^-1=1/v1
exist vector that v1-v1=v1/v1=1 => true
being stuck at others.

Answer 4

why I don't see "specific rule" on the original problem?? I just see specific rule on scalar multiplication only?:

Answer 5

v is vector space if it satisfied these axioms
http://prntscr.com/40md3n

Answer 6

vector addition being ordinary multiplication
that is the specific rule I was talking about
+ become x
x become ^

Answer 7

>.< confused with it

Answer 8

it seems I did it by chance haha
(a + b)v1=av1+bv1=v1^a+v1^b positive real, belongs
a(v1+v2)=a.v1.v2=(v1.v2)^a positive real, belongs

with the 3 I stated before and 1 that @OOPS did, this question was answered :)
thanks guys

Answer 9

wait, have to add that v1^a+v1^b=v1^(a+b) before we can conclude :)