# Understand the Difference Between Trivial and Non-Trivial Solutions

• K3nt70
In summary, the term "trivial" can have different meanings depending on the context. In linear algebra, a trivial solution is simply the zero solution, while in differential equations, it is the identically zero solution. Whether a solution is trivial or not also depends on the initial conditions or the properties of the matrix involved. In some cases, the trivial solution may be the only solution, while in others, there may be an infinite number of 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.

"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.

"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
andrewm said:
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).