prove that a set V is a vector space
Hello,
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...
Re: prove that a set V is a vector space
Quote:
Originally Posted by
WeeG 
Hello,
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...
**x=x^{rs})
++(s**x)=x^rx^s=x^{r+s})
**x\neq(r**x)++(s**x))
So, 
is not a vector space.
If 
, then would be a vector space. Are you sure you've copied the question correctly?
Re: prove that a set V is a vector space
Quote:
Originally Posted by
WeeG 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)
The equality is not correctly written. Take into account that 
is the sum of vectors (not of scalars). You have to prove ^{**}x=r^{**}x++s^{**}x)
. Then,
^{**}x=x^{r+s}=x^rx^s=x^r++x^s=r^{**}x++s^{** }x)
Re: prove that a set V is a vector space
Quote:
Originally Posted by
alexmahone So,
is not a vector space.
Look at my previous post.
Re: prove that a set V is a vector space
Quote:
Originally Posted by
FernandoRevilla
How come you're using 2 different kinds of addition in your equation? The OP's symbol for addition is ++, so why shouldn't the equation be ^{**}x=r^{**}x++s^{**}x)
?
Re: prove that a set V is a vector space
Re: prove that a set V is a vector space
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
Re: prove that a set V is a vector space
Quote:
Originally Posted by
WeeG 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
++ is between two members of V.
** is between a scalar and a member of V.
Re: prove that a set V is a vector space
yes, and what happen when I have 2 scalars, do I use the "regular" operators ?
Re: prove that a set V is a vector space
Quote:
Originally Posted by
WeeG yes, and what happen when I have 2 scalars, do I use the "regular" operators ?
Yes, notice that the problem says and the field is 
without mentioning "strange operations", so the interpretation must be the usual operations.
Re: prove that a set V is a vector space
Quote:
Originally Posted by
WeeG Hello,
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...
How can V as defined above be a vector space? The zero vector is missing from set V...
Have i misunderstood the question?
(Wondering)
Re: prove that a set V is a vector space
Re: prove that a set V is a vector space
Quote:
Originally Posted by
agentmulder How can V as defined above be a vector space? The zero vector is missing from set V...
Have i misunderstood the question?
(Wondering)
I think the zero vector in this case is 
because 
and 
.
Re: prove that a set V is a vector space
Quote:
Originally Posted by
alexmahone I think the zero vector in this case is
because
and
.
suppose x is 2...then 2++1=2?
oh wait...i see now... x++y=xy so 2++1=(2)(1)=2
Re: prove that a set V is a vector space
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?
(Itwasntme)
