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October 20th, 2013, 03:47 AM   #1
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Homogenous system with more variables than equations

Did I solve this correctly?

Solve the system.
3u - v + w -5x - y = 0
6u - 2v + 2w - 9x + y = 0
-9u + 3v - 3w + 11x -y = 0

Augmented matrix:


My answer:
u - 1/3v + 1/3 w = 0
u = 1/3(v - w)
where v and w are any real numbers.

v = 3u + w, where u and w are any real numbers.

w = v - 3u, where v and u are any real numbers.

I was told that I should set v and w to any arbitrary variable and then solve. Such as, let v = p and w = q, where p and q are any real numbers. I agree that this is one way to solve the system, but I don't see how my solution does not.

Any suggestions?
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October 20th, 2013, 03:52 AM   #2
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Re: Homogenous system with more variables than equations

Edit: My answer also included x = 0 and y = 0.
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October 20th, 2013, 02:57 PM   #3
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Re: Homogenous system with more variables than equations

You are writing u , v , and w in terms of each other so it looks 'circular'. Since v and w are free variables the convention is to rename them , this will give

v = p

w = q

u = 1/3(p - q)

x = 0

y = 0

Now , make arbitrary choices for p and q to get u , v , w , x , y to avoid the circular reasoning.

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October 20th, 2013, 04:58 PM   #4
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Re: Homogenous system with more variables than equations

Thanks, that makes sense now. Guess I have to admit it: I was wrong.
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October 22nd, 2013, 02:27 AM   #5
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Re: Homogenous system with more variables than equations

Quote:
Originally Posted by nicnicman
Thanks, that makes sense now. Guess I have to admit it: I was wrong.
Don't worry buddy , the first 4.5 billion years are tough ... thankfully they have passed already ... now it should be 'smooth sailing' from here on forward.

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