Understand the Difference Between Trivial and Non-Trivial Solutions

[K3nt70]

A "trivial" question

I was hoping that somebody could help me understand the difference between trivial and non-trivial solutions. I need to complete some true and false questions for an assignment. For example: If the system is homogeneous, every solution is trivial.

 

[andrewm]

"Trivial", in this context, implies that the solution vector to the system has each component zero.

For instance, Ax=b, where A is NxN and x,b are N-vectors has solutions $$x = inv(A) b$$ and x = 0 when A is invertible, but only x = 0 when A is singular.

So, x = 0 is the trivial solution. It is the only solution when A is singular.

Wikipedia has a description at Trivial_%28mathematics%29.

 

[HallsofIvy]

"trivial" depends upon exactly what you are talking about. Since you refer to "homogeneous systems", I assume you are talking about either Linear Algebra or Linear Differential Equations. In differential equations, a "trivial" solution is the identically zero solution, f(t)= 0 for all t. In Linear Algebra, a "trivial" solution is just the zero solution, x= 0.

It is easy to prove that a system of linear homogeneous differential equations, with a given initial value condition, has a unique solution. It is almost "trivial" (pun intended) to show that the "trivial solution" y= 0 for all x is a solution to every linear homogeneous differential equation. Finally, if the initial value condition is itself "homogeneous", that is, every function is 0 at some initial value of t, y= 0 is the only solution.

Note that that is NOT what you said. Given an initial value condition there is only one solution which- if the initial value condition is homogeneous, is the trivial solution. If you have only a homogenous system of linear differential equations with no initial condition, the trivial solution is one solution but there are an infinite number of non-trivial (i.e. not identically 0) solutions. In neither condition would I say that "{every solution is trivial". Either there is a single, trivial, solution or there exist an infinite number of non-trivial solutions.

In terms of Linear Algebra, a matrix equation (which may be derived from a system of linear equations) of the form Ax= 0 obviously has the "trivial" solution x= 0. If A has an inverse matrix (i.e. if it not singular) then that trivial solution is the only solution. If A is singular then there are an infinite number of non-trivial solutions. Again, in neither case would I say "every solution is is trivial".

I'm not sure what andrewm intended but it is NOT true that

Ax=b, where A is NxN and x,b are N-vectors has solutions x= A^-1b and x = 0 when A is invertible, but only x = 0 when A is singular.

Obviously A0= 0, not b, whether A is singular or not.

What is true is that the equation Ax= 0 have the (trivial) solution x= 0 for any A. It is the only solution if A is NOT singular and there are an infinite number of non-trivial solutions if A is singular.

The equation Ax= b has the unique solution x= A^-1b if A is non-singular. If A is singular, then Ax= b has either no solutions (if b is not in the range of A) or an infinite number of solutions (if b is in the range of A).