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?
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$\begingroup$From an abstract algebra point of view, the best way to understand what trivial is would be to look at situations or examples where it is mostly used and encountered.
Take the case of subsets of a set, say $A$. Since every set of is a subset of itself, $A$ is a trivial subset of itself.
Another situation would be the case of a subgroup. The subset containing only the identity of a group is a group and it is called trivial.
Take a completely different situation. Take the case of a system of linear equations, $$a_1x+b_1y=0\\a_3x+b_4y=0\\a_5x+b_6y=0$$ It is obvious that $x=y=0$ is a solution of such a system of equations. This solution would be called trivial.
Take matrices, if the square of a matrix, say that of $A$, is $O$, we have$A^2=O$. An obvious (trivial) solution would be $A=O$. However, there exist other (non-trivial) solutions to this equation. All non-zero nilpotent matrices would serve as non-trivial solutions of this matrix equation.
$\endgroup$ $\begingroup$I think of the trivial case as the case that invariably arises as the problem’s parameters vary.
In particular, the trivial solution of a homogeneous system of linear equations is the zero vector.
$\endgroup$ $\begingroup$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).
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