Let G?L(R^n;R^n) be the subset of invertible linear transformations. a) For H?L(

Question

Let G?L(R^n;R^n) be the subset of invertible linear transformations. a) For H?L(R^n;R^n), prove that it ||H||<1, then the partial sum L_n=?(k=0,n)H^k converges to a limit L and ||L||?1/(1?||H||). b) If A?L(R^n;R^n) satisfies ||A?I||<1, then A is invertible and A^(?1)=?(k=0,?)H_k where I?A=H. (Hint: Show that AL_n=H^(n+1)) c) Let ?:G?G be the inversion map ?(A)=A^(?1). Prove that ? is continuous at the identity I, using the previous two facts. d) Let A,C?G and B=A^(?1). We can write C=A?K and ?(A?K)=c^(?1)=A^(?1)(I?H)^(?1) where H=BK. Use this to prove that ? is continuous at A.

Explanation / Answer

ANSWER Let G?L(Rn;Rn) be the subset of invertible linear transformations. a) For H?L(Rn;Rn), prove that if ||H||<1, then the partial sum Ln=?nk=0Hk converges to a limit L and ||L||?11?||H||. b) If A?L(Rn;Rn) satisfies ||A?I||<1, then A is invertible and A?1=??k=0Hk where I?A=H. (Hint: Show that ALn=Hn+1) c) Let ?:G?G be the inversion map ?(A)=A?1. Prove that ? is continuous at the identity I, using the previous two facts. d) Let A,C?G and B=A?1. We can write C=A?K and ?(A?K)=c?1=A?1(I?H)?1 where H=BK. Use this to prove that ? is continuous at A. I have little ideas about these questions. What's your answers?

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