一個非齊次線性方程的證明題

I. If x(t), y(t) satisfy x'=A(t)x+f(t), then (x-y)'=A(t)(x-y).
so that, x2(t)-x1(t), x3(t)-x2(t), ..., x(n+1)(t)-xn(t) are sol. of x'=A(t)x
II. x1(t), ..., x(n+1)(t) are linearly indep.
then x2(t)-x1(t), ..., x(n+1)(t)-xn(t) are linearly indep.
III. x1(t) is a particular sol. of x'=A(t)x+f(t), where A(t) is a nxn matrix,
so that, the general sol. of x'=A(t)x+f(t) is
x(t)= particular sol. + general homogeneous sol.
= x1(t) +a(1)[x2(t)-x1(t)]+a(2)[x3(t)-x2(t)]+...+a(n)[x(n+1)(t)-xn(t)]
= [1-a(1)]x1(t)+[a(1)-a(2)]x2(t)+...+[a(n-1)-a(n)]xn(t)+ a(n)x(n+1)(t)
= c(1) x1(t)+c(2) x2(t)+...+ c(n) xn(t)+ c(n+1) x(n+1)(t)
where c(1)+c(2)+...+c(n+1)
=[1-a(1)]+[a(1)-a(2)]+[a(2)-a(3)]+...+[a(n-1)-a(n)] + a(n)
= 1