Matrices - How to prove a sheaf of planes Watch

DeadManProp
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Greetings,

The three equations below form a sheaf of planes and meet along a line. I cannot figure out, however, how to show this algebraically. Does anyone have any ideas on how to manipulate it in order to prove it?

5x + 4y + z = -4
3x + y + 2z = 6
x - y + 2z = 10

Thank you in advance for an explanation.
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Gent2324
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solve for x y and z. there will be an infinite amount of solutions
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DeadManProp
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My problem is I don't know how to show it's a sheaf rather than a triangular prism. i.e. That the equations are consistent.
Last edited by DeadManProp; 4 weeks ago
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mqb2766
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(Original post by DeadManProp)
My problem is I don't know how to show it's a sheaf rather than a triangular prism. i.e. That the equations are consistent.
The equations would be inconsistent in this case.
You want to show theyre consistent.
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RDKGames
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(Original post by DeadManProp)
Greetings,

The three equations below form a sheaf of planes and meet along a line. I cannot figure out, however, how to show this algebraically. Does anyone have any ideas on how to manipulate it in order to prove it?

5x + 4y + z = -4
3x + y + 2z = 6
x - y + 2z = 10

Thank you in advance for an explanation.
Rearrange last eqn for y and sub it into the first two eqns. Try to solve those two equations simultanously. You would find that there are infinitely many solutions.

Hence we have a sheaf.


Alternatively, you can use knowledge of echelon form and gaussian elimination to show this.
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DeadManProp
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(Original post by mqb2766)
The equations would be inconsistent in this case.
You want to show theyre consistent.
According to this 3D calculator and the answer in my textbook, they form a sheaf. Therefore they are consistent unless I'm misunderstanding something.
https://www.geogebra.org/3d

I just don't know how to show, algebraically, that it is the case indeed.
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mqb2766
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basically do row reduction (on the augmented matrix) and the last row should represent
0 = 0
to be consistent. If they were inconsistent, you'd have domething like
0 = 1
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DeadManProp
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(Original post by RDKGames)
Rearrange last eqn for y and sub it into the first two eqns. Try to solve those two equations simultanously. You would find that there are infinitely many solutions.

Hence we have a sheaf.


Alternatively, you can use knowledge of echelon form and gaussian elimination to show this.
Oh wow, thank you! This totally makes sense now :]
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