Vector Spaces

  • Thread starter hkus10
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  • #1
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Suppose that the solution set to a linear system Ax = b is a plane
in R^n with vector equation x = p + su + tv , s, t ∈ R . Prove that
p is a solution to the nonhomogeneous system Ax = b , and that
u and v are both solutions to the homogeneous system Ax = 0 .
(Hint Try choices of s and t).

Should I start from A(p + su + tv) = b? If yes, what should I do from here? If no, where should I start?
 

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  • #2
LCKurtz
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Suppose that the solution set to a linear system Ax = b is a plane
in R^n with vector equation x = p + su + tv , s, t ∈ R . Prove that
p is a solution to the nonhomogeneous system Ax = b , and that
u and v are both solutions to the homogeneous system Ax = 0 .
(Hint Try choices of s and t).

Should I start from A(p + su + tv) = b? If yes, what should I do from here? If no, where should I start?

Yes. And follow the hint.
 
  • #3
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Yes. And follow the hint.
Ap + A(su) + A(tv) = b
Ap + s(Au) + t(Av) = b
Ap + s(0) + t(0) = b
Ap = b

Is this correct?
 
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  • #4
LCKurtz
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what should I go from here?

Sorry. No more help from here until you show us what you have tried following the hint. Show us your effort.

[Edit] Your post hadn't shown up when I wrote this. See my next post.
 
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  • #5
LCKurtz
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Ap + A(su) + A(tv) = b
Ap + s(Au) + t(Av) = b
Ap + s(0) + t(0) = b
Ap = b

Is this correct?

Yes. You have now shown that p is a solution to the NH equation. Now try something else along those lines...
 
  • #6
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Now try something else along those lines...
What is the goal for that?
 
  • #7
LCKurtz
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Suppose that the solution set to a linear system Ax = b is a plane
in R^n with vector equation x = p + su + tv , s, t ∈ R . Prove that
p is a solution to the nonhomogeneous system Ax = b , and that
u and v are both solutions to the homogeneous system Ax = 0 .
(Hint Try choices of s and t).


Should I start from A(p + su + tv) = b? If yes, what should I do from here? If no, where should I start?

Yes. You have now shown that p is a solution to the NH equation. Now try something else along those lines...

What is the goal for that?

Because you aren't done. See the red above.
 

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