What does "multiple non-trivial solutions exists mean?"

Question

I came across this question of solving system of linear non-homogeneous equations :

$$x+2y+z+4w=2$$ $$3x+6y+3z+12w=6$$

Options are :

1. Only the trivial solution $x=y=z=w=0$ exists
2. There is no solution
3. A unique non-solution exists
4. Multiple non-trivial solutions exists.

The answer is Multiple non-trivial solutions exists.

Can somebody explain me why and what does it mean? I always thought the rows in the matrix must be equal to the number of variables?

Answer 1

"Multiple non-trivial solutions exist": a solution is called nontrivial if it is not identically zero (like in your option 1). So this statement means there are at least two different solutions to that equation which are not that particular zero solution.

Edit (actually the trivial solution does not satisfy the equation(s), so it is not a solution).

Answer 2

You may have to consider systems of linear equations for which the number of equations is different from the number of unknowns.

For $m\times n$ homogeneous systems (right-hand side is $0$), there is at least the trivial solution. The set of solutions is a subspace of $\mathbf R^n$. If the system of equations has rank $r$, the space of solutions has codimension $r$, i.e. it has dimension $n-r$.

For $m\times n$ non-homogeneous systems, there is a solution if and only if the rank of the augmented matrix is equal to the rank of the matrix of the homogeneous system. In such cases, the set of solutions is an affine space directed by the vector space of solutions of the homogeneous associated system.