Final answer:
Trivial solutions in linear algebra refer to setting all variables to zero in a homogeneous system, while nontrivial solutions involve at least one variable not being zero, indicating multiple solutions.
Step-by-step explanation:
The difference between trivial and nontrivial solutions in linear algebra relates to the solutions of a homogeneous system of linear equations. A trivial solution is one in which all the variables are set to zero. This solution always exists for a homogeneous system. In contrast, a nontrivial solution is one where at least one variable is not equal to zero, indicating that there are infinite solutions to the system other than the trivial solution.
For example, consider the system of equations:
The trivial solution would be x = 0, y = 0. A nontrivial solution might be x = 1, y = -1, which also satisfies both equations. Whether nontrivial solutions exist depends on the system's coefficients and its rank.