Is the trivial solution a unique solution?

  • Thread starter Thread starter darthxepher
  • Start date Start date
Click For Summary
If the only solution to the equation Ax = b is the trivial solution (x=0), then it is considered unique if b is zero and A is invertible. A non-zero b means there is no trivial solution. The definition of a unique solution is that if it is the only solution, it is unique regardless of its triviality. The discussion also briefly touches on least squares approximations, noting that they always exist, although their relevance to unique solutions is unclear. The focus remains on the conditions under which the trivial solution is deemed unique.
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.
 

Thread 'Confusion about the Moyal-Weyl twist'

I am studying the mathematical formalism behind non-commutative geometry approach to quantum gravity. I was reading about Hopf algebras and their Drinfeld twist with a specific example of the Moyal-Weyl twist defined as F=exp(-iλ/2θ^(μν)∂_μ⊗∂_ν) where λ is a constant parametar and θ antisymmetric constant tensor. {∂_μ} is the basis of the tangent vector space over the underlying spacetime Now, from my understanding the enveloping algebra which appears in the definition of the Hopf algebra...
View full post »

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
Can a Singular Matrix Have a Unique Solution for Ax=b?
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Uniqueness of reduced row echelon form (proof verification)
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Clarifications needed for an exercise in Hungerford's abstract algebra
  • · Replies 23 ·
Replies
23
Views
953
MHB Is G/G isomorphic to the trivial group? A proof for G/G\cong \{e\}
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad True or False: A Linear System Must Have a Unique Solution
  • · Replies 1 ·
Replies
1
Views
1K
MHB Matrices- conditions for unique and no solution
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Can a group be isomorphic to one of its quotients?
  • · Replies 4 ·
Replies
4
Views
2K
MHB 311.1.3.19 find h and k in the system
  • · Replies 1 ·
Replies
1
Views
1K