Is the trivial solution a unique solution?

  • Context: Undergrad 
  • Thread starter Thread starter darthxepher
  • Start date Start date
Click For Summary

Discussion Overview

The discussion centers around the uniqueness of the trivial solution in the context of linear equations of the form Ax = b. Participants explore conditions under which the trivial solution is considered unique, particularly focusing on the cases when b is zero or non-zero, and the implications of matrix A being invertible.

Discussion Character

  • Debate/contested

Main Points Raised

  • Some participants assert that if the only solution to Ax = b is the trivial solution, then it is considered unique by definition.
  • Others clarify that if b is non-zero, there cannot be a trivial solution, while if b is zero and A is invertible, then x=0 is indeed the unique solution.
  • One participant questions the relevance of least squares approximation to the topic of unique solutions, suggesting that approximations exist regardless of their quality.

Areas of Agreement / Disagreement

There is no consensus on the relationship between trivial solutions and uniqueness, as participants present differing views on the conditions under which a trivial solution can be considered unique.

Contextual Notes

The discussion does not resolve the implications of non-zero b on the existence of trivial solutions, nor does it clarify the connection between least squares approximation and unique solutions.

darthxepher
Messages
56
Reaction score
0
If the only solution to Ax = b is the trivial solution, then is that solution considered unique?
 
Physics news on Phys.org
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
 
Do you know how to tell if the least squares approximation exists?
 
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.
 
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.
 

Similar threads

Undergrad Existence and Uniqueness of Inverses
  • · Replies 5 ·
Replies
5
Views
2K
Graduate Matrix Multiplication: Solving A*A'=B for Unique A(B)
  • · Replies 18 ·
Replies
18
Views
2K
Undergrad Uniqueness of reduced row echelon form (proof verification)
  • · Replies 4 ·
Replies
4
Views
3K
Graduate Can a Singular Matrix Have a Unique Solution for Ax=b?
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad True or False: A Linear System Must Have a Unique Solution
  • · Replies 1 ·
Replies
1
Views
1K
MHB Is G/G isomorphic to the trivial group? A proof for G/G\cong \{e\}
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Clarifications needed for an exercise in Hungerford's abstract algebra
  • · Replies 23 ·
Replies
23
Views
2K
High School System of linear equations: When are we guaranteed a unique solution?
  • · Replies 5 ·
Replies
5
Views
2K
MHB Matrices- conditions for unique and no solution
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Can one find a matrix that's 'unique' to a collection of eigenvectors?
  • · Replies 33 ·
2
Replies
33
Views
3K