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Linear Algebra Proof

  • Thread starter dmitriylm
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  • #1
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Homework Statement



Prove that if a system with rational coeffcients and constants has a solution then it has at least one all-rational solution. If such as system has infinitely many solutions, will it also have infinitely many all-rational solutions ?


Homework Equations





The Attempt at a Solution



So I'm taking this Linear Algebra course and I've never had such a hard time answering what appear to be very simple questions (and I had no issues with calc 1 / calc 2!). I understand that in linear algebra there is either one solution, no solutions, or infinitely many solutions. These are the only three possible outcomes. Where do I go from there? I would greatly appreciate any help/guidance.
 

Answers and Replies

  • #2
Dick
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Generally the way to start thinking about problems like this is to pick a simple example and think about it. Take one equation in two unknowns, like r1*x+r2*y=r3 where r1, r2, and r3 are rational. That generally has an infinite number of solutions. Can you say why it has rational solutions?
 
  • #3
lanedance
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so start with Ax =b, with A a matrix & b a vector, each with rational components what can you say if the system has a solution?
 
  • #4
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Generally the way to start thinking about problems like this is to pick a simple example and think about it. Take one equation in two unknowns, like r1*x+r2*y=r3 where r1, r2, and r3 are rational. That generally has an infinite number of solutions. Can you say why it has rational solutions?
Because the solutions have to equal a rational number?
 
  • #5
Dick
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Because the solutions have to equal a rational number?
Well, no! They don't HAVE to be rational. If x=sqrt(2) the solution isn't rational, is it? The question just asks if there IS a rational solution. You should keep thinking about this.
 
  • #6
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Well, no! They don't HAVE to be rational. If x=sqrt(2) the solution isn't rational, is it? The question just asks if there IS a rational solution. You should keep thinking about this.
Is it because with an equation with an infinite number of solutions, at least one of those solutions must be a rational number?
 
  • #7
Dick
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Is it because with an equation with an infinite number of solutions, at least one of those solutions must be a rational number?
Not just any equations. You have to figure out what kind of equations you get from solving the system with rational coefficients. Then SHOW it has a rational solution. Not just say it does.
 

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