by the way are the Marsden texts good? it seems some people like Spivack and Courant because they introduce manifolds earlier
Show that the system of equations
X1 = (1/4)x1- (1/4)x2 +(2/15)x3 +3
X2=(1/4)x1 + (1/5)x2 + (1/2)x3-1
X3 = (-1/4)x1+(1/3)x2-(1/3)x3+2
Has a unique solution, using the contraction mapping principle. (Hint: Choose a clever norm on R3 or estimate using the Schwarz inequality)