the fundamental matrix
Once you have a handful of building-block solutions, it is natural to file them side by side into one tidy object instead of carrying them around as a loose list. Stand each solution vector up as a column of a matrix, and you get the fundamental matrix — a single matrix-valued function of t that contains the whole solution set at once.
Given a fundamental set x_1(t), ..., x_n(t) of a homogeneous system x' = A(t) x, the fundamental matrix is Psi(t) = [x_1(t) | x_2(t) | ... | x_n(t)], whose columns are those solutions. Because each column is a solution, the matrix itself satisfies the same equation upgraded to matrix form: Psi'(t) = A(t) Psi(t). And because the columns are independent, Psi(t) is invertible at every t — its determinant is the Wronskian, which is nonzero. The payoff is that the general solution is just x(t) = Psi(t) c, where c is a constant column vector of coefficients; choosing c selects which solution you mean. For an initial condition x(t0) = x0, you solve Psi(t0) c = x0 for c, giving x(t) = Psi(t) Psi(t0)^(-1) x0.
The fundamental matrix is the workhorse object of linear-system theory. It packages superposition into matrix multiplication, it makes the initial-value problem a single inverse-and-multiply, and it is the bridge to the matrix exponential: for constant-coefficient systems the particular fundamental matrix normalized to the identity at t0 is exactly e^(A t).
For x' = [1, 0; 0, -2] x, the columns (e^t, 0) and (0, e^(-2t)) give Psi(t) = [e^t, 0; 0, e^(-2t)]. Then x(t) = Psi(t) c reproduces every solution, and x(0) = (3, 5) forces c = (3, 5), so x(t) = (3 e^t, 5 e^(-2t)).
Columns are the building-block solutions; multiplying by the constant vector c assembles them into one particular motion.
A fundamental matrix is not unique — any choice of fundamental set gives a valid Psi, and any Psi times a constant invertible matrix is another valid one. The one normalized so that Psi(t0) = I is special and called the principal fundamental matrix.