What is a trivial and a non-trivial solution in terms of linear algebra? The homogeneous unique solution always gives a trivial solution but trivial solution consists of zeros e.g $\{0,0,0\}$ but what if the solution is still homogeneous unique solution but not in the form of zeros e.g $\{1,2,3\}$.   
Is this solution also trivial? How?
 A: Trivial solution is a technical term. For example, for the homogeneous
linear equation $7x+3y-10z=0$ it might be  a trivial affair  to find/verify that  $(1,1,1)$ is a solution. But the term trivial solution is reserved  exclusively for for the solution consisting of zero values for all the variables.
There are similar trivial things in other topics. Trivial group is one that consists of just one element, the identity element.  Trivial vector bundle is  actual product with vector space (instead of one that is merely looks like a product locally over  sets in an open covering).
Warning  in non-linear algebra this is used in different meaning. Fermat's theorem dealing with polynomial equations of higher degrees states that for $n>2$, the equation $X^n+Y^n=Z^n$ has only trivial solutions for integers $X,Y,Z$. Here trivial refers to besides  the trivial trivial one $(0,0,0)$ the next trivial ones $(1,0,1), (0,1,1)$ and their negatives  for even $n$.
A: A homogeneous system will always have the zero vector as a solution. So what you ask can never happen. 
