Is the trivial solution a unique solution?

  • Thread starter darthxepher
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In summary, if the only solution to Ax = b is the trivial solution, then it is considered unique. This is true regardless of whether b is non-zero or zero, as long as A is invertible. Additionally, the existence of a least squares approximation is unrelated to the uniqueness of the solution.
  • #1
darthxepher
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If the only solution to Ax = b is the trivial solution, then is that solution considered unique?
 
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  • #2
If b is non-zero, there is no trivial solution.

If b is zero, and A is invertible, then yes, x=0 is the unique solution
 
  • #3
Do you know how to tell if the least squares approximation exists?
 
  • #4
darthxepher said:
If the only solution to Ax = b is the trivial solution, then is that solution considered unique?

If it's the only solution, then by definition it's unique.
 
  • #5
It doesn't matter whether it is trivial or not, "only solution"= "unique solution".

I have no idea what "Do you know how to tell if the least squares approximation exists?" has to do with "unique solutions" but approximations, whether good or bad, always exist.
 

1. What is the difference between a trivial solution and a unique solution?

A trivial solution is a solution that is always true, regardless of the variables or inputs. A unique solution is a solution that is the only possible answer to a given problem.

2. Can a problem have both a trivial solution and a unique solution?

No, a problem can only have one type of solution. If a problem has a unique solution, it cannot also have a trivial solution.

3. How can you determine if a solution is trivial or unique?

To determine if a solution is trivial or unique, you must first define the problem and the variables involved. Then, you can plug in the values of the variables and see if the solution always holds true or if it is the only possible answer.

4. Is the trivial solution always the simplest solution?

No, the trivial solution is not always the simplest solution. While it may be simple in its form, it may not be the most efficient or effective solution for a given problem.

5. Are trivial solutions useful in scientific research?

Trivial solutions can be useful in scientific research as they provide a starting point for more complex problems. They can also help researchers understand the relationships between variables and identify patterns in data.

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