402 CHAPTER 16. VECTOR SPACES

28. ↑Show there exists a special Φ such that Φ′ (t) = AΦ(t) , Φ(0) = I, and Φ(t)−1

exists for all t. Show using uniqueness that

Φ(−t) = Φ(t)−1

and that for all t,s ∈ RΦ(t + s) = Φ(t)Φ(s)

Explain why with this special Φ, the solution to 16.13 can be written as

x(t) = Φ(t− t0)x0 +∫ t

t0Φ(t− s) f(s)ds.

Hint: Let Φ(t) be such that the jth column is x j (t) where

x′j = Ax j, x j (0) = e j.

Use uniqueness as required.

29. ∗Using the Lindemann Weierstrass theorem show that if σ is an algebraic numbersinσ ,cosσ , lnσ , and e are all transcendental. Hint: Observe, that

ee−1 +(−1)e0 = 0, 1eln(σ)+(−1)σe0 = 0,

12i

eiσ − 12i

e−iσ +(−1)sin(σ)e0 = 0.