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Math Help - prove that a set V is a vector space

  1. #1
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    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...
    Last edited by WeeG; December 19th 2011 at 11:26 PM.
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  2. #2
    MHF Contributor alexmahone's Avatar
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    Re: prove that a set V is a vector space

    Quote Originally Posted by WeeG View Post
    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...
    (r++s)**x=x^{rs}

    (r**x)++(s**x)=x^rx^s=x^{r+s}

    (r++s)**x\neq(r**x)++(s**x)

    So, V is not a vector space.

    If a**x=a^x, then V would be a vector space. Are you sure you've copied the question correctly?
    Last edited by alexmahone; December 19th 2011 at 11:42 PM.
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  3. #3
    MHF Contributor FernandoRevilla's Avatar
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    Re: prove that a set V is a vector space

    Quote Originally Posted by WeeG View Post
    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 (r+s)^{**}x=r^{**}x++s^{**}x . Then,


    (r+s)^{**}x=x^{r+s}=x^rx^s=x^r++x^s=r^{**}x++s^{**  }x
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    MHF Contributor FernandoRevilla's Avatar
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    Re: prove that a set V is a vector space

    Quote Originally Posted by alexmahone View Post
    So, V is not a vector space.
    Look at my previous post.
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  5. #5
    MHF Contributor alexmahone's Avatar
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    Re: prove that a set V is a vector space

    Quote Originally Posted by FernandoRevilla View Post
    (r+s)^{**}x=r^{**}x++s^{**}x
    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 (r++s)^{**}x=r^{**}x++s^{**}x?
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  6. #6
    MHF Contributor FernandoRevilla's Avatar
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    Re: prove that a set V is a vector space

    Quote Originally Posted by alexmahone View Post
    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 (r++s)^{**}x=r^{**}x++s^{**}x?
    Let us see, the field is (\mathbb{R},+,\cdot) with the usual operations sum and product. The operation defined on V is ++ . So, for two scalars r,s its sum is r+s , not r++s .
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  7. #7
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    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
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  8. #8
    MHF Contributor alexmahone's Avatar
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    Re: prove that a set V is a vector space

    Quote Originally Posted by WeeG View Post
    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.
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  9. #9
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    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 ?
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    MHF Contributor FernandoRevilla's Avatar
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    Re: prove that a set V is a vector space

    Quote Originally Posted by WeeG View Post
    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 \mathbb{R} without mentioning "strange operations", so the interpretation must be the usual operations.
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    Member agentmulder's Avatar
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    Re: prove that a set V is a vector space

    Quote Originally Posted by WeeG View Post
    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?

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  12. #12
    MHF Contributor FernandoRevilla's Avatar
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    Re: prove that a set V is a vector space

    Quote Originally Posted by agentmulder View Post
    How can V as defined above be a vector space? The zero vector is missing from set V... Have i misunderstood the question?
    For all x\in V we have x++1=1++x=x so, 1 is the zero vector of (V,++) .
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  13. #13
    MHF Contributor alexmahone's Avatar
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    Re: prove that a set V is a vector space

    Quote Originally Posted by agentmulder View Post
    How can V as defined above be a vector space? The zero vector is missing from set V...

    Have i misunderstood the question?

    I think the zero vector in this case is 1 because x++1=x and a**1=1.
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  14. #14
    Member agentmulder's Avatar
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    Re: prove that a set V is a vector space

    Quote Originally Posted by alexmahone View Post
    I think the zero vector in this case is 1 because x++1=x and a**1=1.
    suppose x is 2...then 2++1=2?

    oh wait...i see now... x++y=xy so 2++1=(2)(1)=2
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  15. #15
    Member agentmulder's Avatar
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    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?

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