Q: For a system of 3 homogeneous linear equations in 3 variables, is it compulsory for a (0,0,0) solution to be unique in order to be called trivial?
In other words do the following equations have a trivial solution or non trivial solution or both? 2x+y+z=0 4x+2y+2z=0 8x+4y+4z=0
If the answer is 'both' then what's the difference between a zero solution and trivial solution?
Without even looking at the question, we know that $(0,0,0)$ is a valid solution, that is why it is the trivial solution regardless of whether there is any other solution.
If there is any non-zero solution, we call it a non-trivial solution.
Every homogeneous system has the trivial solution as a solution, the question of interest is whether there is any non-trivial ones.