Differential equations. linear system.

[dictation]

Homework Statement

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).

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..

 

[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 x_n(t) defined as the solution with x_n(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, ..., a_n) is equal to a₁x₁(t)+ a₂x₂(t)+ a₃x₃(t)+ ...+ a_nx_n(t) by showing that they both satisfy the differential equation and the same initial condition.