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True/False Question on Linear Combos

  • Thread starter NullSpace0
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  • #1
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Homework Statement


If the system Ax=b has two known solutions, x1 and x2, true or false that another solution exists. If it does, find another solution.


Homework Equations





The Attempt at a Solution


I think it's true. But after that, I don't know.

I'm really lost, but I think I remember learning that given two solutions, any linear combination of the two is also a solution. Does that mean that x1+x2 is a solution?

I'm worried that that is actually a solution to Ax=2b, though.
 

Answers and Replies

  • #2
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Perhaps that is only true for homogeneous equations, now that I think about it.

Would nonhomogeneous equations require some sort of weighted average linear combination?
 
  • #3
SammyS
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Homework Statement


If the system Ax=b has two known solutions, x1 and x2, true or false that another solution exists. If it does, find another solution.

Homework Equations



The Attempt at a Solution


I think it's true. But after that, I don't know.

I'm really lost, but I think I remember learning that given two solutions, any linear combination of the two is also a solution. Does that mean that x1+x2 is a solution?

I'm worried that that is actually a solution to Ax=2b, though.
Is it true that (1/2)Ax = A((1/2)x) ?
 
  • #4
LCKurtz
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Homework Statement


If the system Ax=b has two known solutions, x1 and x2, true or false that another solution exists. If it does, find another solution.

.
Is it given b is not the zero vector? Think about uniqueness, rank, and free variables.
 

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