Linear Algebra Proof: Prove if Rational Solutions Exist

[dmitriylm]

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.

 

[Dick]

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?

 

[lanedance]

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?

 

[dmitriylm]

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?

 

[Dick]

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.

 

[dmitriylm]

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?

 

[Dick]

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.