What is meant by "nontrivial solution"? In a linear algebra book, I find the term "nontrivial solution" and can not understand what that means. Could someone please explain what this means?

Suppose we want to find $x_1$ and $x_2$ such that:
$$a_1x_1+a_2x_2=0$$
Well, we could pick $x_1=x_2=0$: This is called the trivial solution of the linear equation.

Now, suppose we want to find $x_1$,$x_2$,...,$x_n$ such that:
$$a_1x_1+a_2x_2+....+a_nx_n=0$$
Well, we could pick $x_1=x_2=...=x_n=0$: This is called the trivial solution of the linear equation.

Now, suppose we want to find $x_1$,$x_2$,...,$x_n$ such that:
$$a_{1,1}x_1+a_{1,2}x_2+....+a_{1,n}x_n=0$$
$$a_{2,1}x_1+a_{2,2}x_2+....+a_{2,n}x_n=0$$
.......
$$a_{n,1}x_1+a_{n,2}x_2+....+a_{n,n}x_n=0$$
Well, we could pick $x_1=x_2=...=x_n=0$: This is called the trivial solution of the homogenous system of linear equations. (homogeneous because the right side of all the equations is zero).
