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Thread: proof of no existence of liner system over R

  1. #1
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    proof of no existence of liner system over R

    Hi, I hope I;m writing this correctly:
    I need to proof that there is no existence liner system over R that this is the set of the answers:
    \left \{ \left ( a,a^{2},b \right )\mid a,b \in \mathbb{R}\right \}
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    Re: proof of no existence of liner system over R

    Quote Originally Posted by xbox360 View Post
    Hi, I hope I;m writing this correctly:
    I need to proof that there is no existence liner system over R that this is the set of the answers:
    \left \{ \left ( a,a^{2},b \right )\mid a,b \in \mathbb{R}\right \}
    Do you mean linear where you wrote liner.
    How does your textbook/notes define linear system?
    Are you asking about a set generating a subspace?

    Please review you post an if necessary correct it or explain it.
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    Re: proof of no existence of liner system over R

    It is obvious that the "linear system" x= a, y= a^2, z= b has the solution set \{a, a^2, b\} so what you say you are trying to prove is not true! Please check your problem again.
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    Re: proof of no existence of liner system over R

    I'm trying to proof that there are no linear system in the field \mathbb{R} that this is the solution set
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    Re: proof of no existence of liner system over R

    Quote Originally Posted by xbox360 View Post
    I'm trying to proof that there are no linear system in the field \mathbb{R} that this is the solution set
    Please answer the questions about this question in replies #2 & #3.
    Can you type out the exact question as it was posed to you?
    Can you give the exact relevant definitions you are using?

    Otherwise, I think reply #3 answers your question.
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  6. #6
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    Re: proof of no existence of liner system over R

    Quote Originally Posted by xbox360 View Post
    I'm trying to proof that there are no linear system in the field \mathbb{R} that this is the solution set
    Then you are trying to prove (not "proof") something that is not true!
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    Re: proof of no existence of liner system over R

    basically, I'm struggling to translate this question, I know that there is no linear equation system in R that this is the solution set of her, I don't know how to prove that.
    Last edited by xbox360; Apr 2nd 2017 at 05:24 AM.
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  8. #8
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    Re: proof of no existence of liner system over R

    Quote Originally Posted by xbox360 View Post
    basically, I'm struggling to translate this question, I know that there is no linear equation system in R that this is the solution set of her, I don't know how to prove that.
    One more time: what you "know" is not true!
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    Re: proof of no existence of liner system over R

    sorry, this is the question I've been asked for, to prove that a system of linear equations doesn't exist over \mathbb{R} for the solutions set \left \{ \left ( a,a^2,b \right )|a,b \in \mathbb{R} \right \}. I can't understand why you find this question wrong.
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  10. #10
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    Re: proof of no existence of liner system over R

    I find this wrong because I know that, as I said in my first response,
    x= a, y= a^2, z= b is precisely such a system of equations!
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    Re: proof of no existence of liner system over R

    Quote Originally Posted by HallsofIvy View Post
    I find this wrong because I know that, as I said in my first response,
    x= a, y= a^2, z= b is precisely such a system of equations!
    Is it possible that she/he means something completely other by linear system?
    Could it be a linear operator?.
    I asked for a working definition be posted, but not get it.
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    Re: proof of no existence of liner system over R

    I think the previous responders didn't understand your question.
    Given a linear system $AX=B$ where $A$ is a 3 by 3 matrix, the solution set $S$ is a subspace of $R^3$ if $B=0$ or the set of differences of members of $S$ is a subspace. So first the set $S=\{(a,a^2,b)\,:a\in R,\,b\in R\}$ is not a subspace:$(0,0,1)\text{ and }(1,1,0)\in S$, but the difference $(0,0,1)-(1,1,0)=(-1,-1,1)\not\in S$ since there is no $a\in R$ with $a^2=-1$. Next the set of differences $S_1=\{(a_1-a_2,a_1^2-a_2^2,b_1-b_2)\,:\,a_1,b_1,a_2,b_2\in R\}$. Suppose $S_1$ is a subspace. Then $(0,0,1)-(0,0,0)=(0,0,1)\in S_1$, $(1,1,0)-(-1,1,0)=(2,0,0)\in S_1$ and $(1,1,0)-(0,0,0)=(1,1,0)\in S_1$. Clearly these 3 vectors are linearly independent members of $S_1$. So $S_1=R^3$. In particular, $(0,1,0)\in S_1$ or for suitable real values $(0,1,0)=(a_1-a_2,a_1^2-a_2^2,b_1-b_2)$. But this says $a_1=a_2$ and so $a_1^2-a_2^2=1$ is a contradiction. QED.
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    Re: proof of no existence of liner system over R

    Quote Originally Posted by johng View Post
    I think the previous responders didn't understand your question.
    I hope that is just a poorly composed sentence (poor choice of words). I think that both of both of us just wanted the poster to take responsibility for making the question clear. It is of great interest to me, at least, if the posters can understand your reply. I hope she/he will tell us.
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    Re: proof of no existence of liner system over R

    well, I didn't fully understand, mostly because we didn't cover wet subspaces and vector space.
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    Re: proof of no existence of liner system over R

    If you are asked to do a problem like this, then surely you must have been given a definition of "linear system". What was that definition?
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