I am trying to prove that a set V is a vector space, I managed to prove most of the process, but stuck with a couple of things...
The set V is the set of positive real numbers, and the field is R.
the two operations are:
x++y = xy
a**x = x^a
(where ++ is the addition, ** is the scalar multiplication, and a is a real scalar)
I find it hard to prove that for 2 scalars r and s, and a member of V, x
(r++s)**x = (r**x) + (s**x)
basically what I don't understand, is what do to when having
r**s, in other words, when I have to multiply 2 scalars, do I do it using the regular operator, or using the new X*scalar operator
any help would be appreciated...
thank you both !
let me get is straight, you are saying, that the operators ++ and ** are between a scalar and a member of V (even if V is number too), and when dealing with 2 scalars, I use the "regular" operators ?
sounds logical I suppose
let me ask you if my interpretation of a**x = a^x is correct...
suppose 1 plays the role of the 0 vector then a**1 is the zero vector since scalar multiplication on the 0 vector gives the zero vector.
now, 1^a = 1 for real number scaler a and real number 1
since 1 is the 0 vector...it checks out
is this correct?
but somehow the vectors are being mixed up with the scalers... or am i thinking too much?