a fundamental set of vector solutions
Imagine you want to describe every possible motion of a system without listing infinitely many of them. The trick is the same as describing every point in a plane using just two arrows: pick a few well-chosen building-block solutions, and write all the rest as combinations of those. The chosen building blocks are a fundamental set.
For an n-dimensional homogeneous linear system x' = A(t) x, a fundamental set is a collection of n solution vectors x_1(t), ..., x_n(t) that are linearly independent — none of them is a combination of the others. The key theorem says these n independent solutions are enough: every single solution of the system can be written as c1 x_1(t) + ... + cn x_n(t) for some constants c1, ..., cn. So the fundamental set is a basis for the entire solution space, and the constants are pinned down by the initial condition. Independence is checked at any one instant t0 using the Wronskian, the determinant of the matrix whose columns are the solution vectors; if it is nonzero there, the set is fundamental.
This is the precise reason 'the general solution' of an n-by-n system is a sum of exactly n terms — not more, not fewer. It mirrors the second-order case, where two independent solutions span everything, generalized to n unknowns. Finding a fundamental set IS solving the homogeneous system; for constant-coefficient A the eigenvalue method delivers one directly.
For x' = [1, 0; 0, -2] x the two solutions (e^t, 0) and (0, e^(-2t)) are independent (their Wronskian is e^t times e^(-2t), never zero), so they form a fundamental set. Every solution is c1 (e^t, 0) + c2 (0, e^(-2t)).
Two independent solutions of a 2-by-2 system span every solution — just as two arrows span a plane.
Independence of vector functions is checked at a single point t0, not pointwise everywhere; for genuine ODE solutions, a nonzero Wronskian at one t0 guarantees it is nonzero for all t (Abel's theorem).