# Homework Help: Differential equations. linear system.

1. Jan 27, 2010

### dictation

1. The problem statement, all variables and given/known data

G(t) is nxn matrix depends on t.
Show that solutions of x'=G(t)x form an n-dim subspace of C1(R+,Rn).

3. The attempt at a solution

So I can show closure, addition of solutions returns some combo inside R^n, and same with scalar multiplication. I need to show dimension..

2. Jan 28, 2010

### HallsofIvy

Since Gn is an n by n matrix, x must be a column matrix with n rows. Let x1(t) be the solution with x(0)1= (1, 0, 0, ..., 0)T. Let x2(t) be the solution with x2(0)= (0, 1, 0, ..., 0)T. Let x3(t) be the solution with x3(0)= (0, 0, 1, ..., 0)T. Continue until you have xn(t) defined as the solution with xn(t)= (0, 0, 0, ..., 1)T. Show that they are independent, by showing that the only solution to the differential equation with x(0)= (0, 0, 0, ..., 0)T is the 0 function, and that the solution with x(t)= (a1, a2, a3, ..., an) is equal to a1x1(t)+ a2x2(t)+ a3x3(t)+ ...+ anxn(t) by showing that they both satisfy the differential equation and the same initial condition.