(For those whove thought about convergence issues) Check that the power series expansion for eA converges for any n n matrix, as follows. (Thinking of the vector spaceMnn of n n matrices as Rn2 makes what follows less mysterious.) a. If A = & aij ' , set _A_ = ( n i,j=1 a2 ij . Prove that (i) _cA_ = |c|_A_ for any scalar c (ii) _A + B_ _A_ + _B_ for any A,B Mnn (iii) _AB_ _A__B_ for any A,B Mnn. (Hint: Express the entries of the matrix product in terms of the row vectors Ai and the column vectors bj .) In particular, deduce that _Ak_ _A_k for all positive integers k. b. It is a fact from analysis that if vk RN is a sequence of vectors in RN with the property that k=1 _vk_ converges (in R), then k=1 vk converges (in RN). Using this fact, prove that k=0 Ak k! converges for any matrix A Mnn. c. (For those who know what a Cauchy sequence is) Prove the fact stated in part b.

L7 - 2 Def. One-Sided Limits We say that a functio fn(x)asm t L as x approaches the numberc from the right if we can make every value of f(x) as close to L as we want by choosing x suﬃciently close to c but x>c . We write thir sight-hand limit...