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

  1. #16
    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
    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?
    Yes. (See post #13.)
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  2. #17
    Member agentmulder's Avatar
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    Re: prove that a set V is a vector space

    Quote Originally Posted by FernandoRevilla View Post
    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
    WOW! this algebraic manipulation between operators is beautiful...
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  3. #18
    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
    but somehow the vectors are being mixed up with the scalers... or am i thinking too much?
    Write V=\{\vec{x},\vec{y},\ldots\} and \mathbb{R}=\{a,b,\ldots\} . Then, \vec{0}=1 , so

    \boxed{a^{**}\vec{0}}=1^a=1=\boxed{\vec{0}}
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  4. #19
    Member agentmulder's Avatar
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    Re: prove that a set V is a vector space

    Quote Originally Posted by FernandoRevilla View Post
    Write V=\{\vec{x},\vec{y},\ldots\} and \mathbb{R}=\{a,b,\ldots\} . Then, \vec{0}=1 , so

    \boxed{a^{**}\vec{0}}=1^a=1=\boxed{\vec{0}}
    yes, thank you, i see. Apparantly the positive real numbers are a vector space according to the operations defined in the OP

    Another question... ++ is regular multiplication, ** is regular exponentiation

    we don't have to worry about the 'object' 0 being in V since no positive base to a real number exponent can give 0.

    I don't know what to call 0... a vector?... a scalar? both?

    Are not the elements of V bases? The elements of R exponents?

    That would mean the bases are vectors the exponents scalars?

    is this correct?

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  5. #20
    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
    I don't know what to call 0... a vector?... a scalar? both?
    That depends. For example, if you consider the "object" 1 as belonging to V , it is a vector, and satisfies 1++x=x++1=x for all x\in V , so 1 is the zero vector. If you consider the "object" 1 as belonging \mathbb{R} , it is a scalar and satisfies a\cdot 1=1\cdot a=a , so 1 is the identity element for the product of scalars.

    On the other hand, the "object" 0 does not belong to V but belongs to \mathbb{R} so, necessarily is an scalar, satisfying a+0=0+a=0 for all a\in\mathbb{R} so, 0 is the identity element for the sum of scalars.
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  6. #21
    Member agentmulder's Avatar
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    Re: prove that a set V is a vector space

    Quote Originally Posted by FernandoRevilla View Post
    That depends. For example, if you consider the "object" 1 as belonging to V , it is a vector, and satisfies 1++x=x++1=x for all x\in V , so 1 is the zero vector. If you consider the "object" 1 as belonging \mathbb{R} , it is a scalar and satisfies a\cdot 1=1\cdot a=a , so 1 is the identity element for the product of scalars.

    On the other hand, the "object" 0 does not belong to V but belongs to \mathbb{R} so, necessarily is an scalar, satisfying a+0=0+a=0 for all a\in\mathbb{R} so, 0 is the identity element for the sum of scalars.
    Woderful! A clear and precise explanation, well done sir.

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