Hal 5. Let CR denote the r Prove curve 1«R for some R21 au alying on Ce.
4. Let A, B CR be non-empty open sets. Prove that AU B is an open set.
4. For A S R. let A denote the interior of A, and A' denote the derived set of A. Prove that A SA'. [5]
*Let . ., A, denote the eigenvalues of an n x n matrix A. Prove that the Frobenius 5. norm of A satisfies ΑIFΣ. i=1 *Let . ., A, denote the eigenvalues of an n x n matrix A. Prove that the Frobenius 5. norm of A satisfies ΑIFΣ. i=1
44. a.Let A and B be two 2 × 2 matrices,Let Tr denote the trace and det denote the determinant. Prove that Tr(AB)-Tr(BA) and det(AB) - det(BA). b. If A is any matrix in SLa(R), prove that det ((-A-t +1 where t = Tr(A). 44. a.Let A and B be two 2 × 2 matrices,Let Tr denote the trace and det denote the determinant. Prove that Tr(AB)-Tr(BA) and det(AB) - det(BA). b. If A is any matrix in SLa(R), prove...
(a) Let R be a commutative ring. Given a finite subset {ai, a2, , an} of R, con- sider the set {rial + r202 + . . . + rnan I ri, r2, . . . , rn є R), which we denote by 〈a1, a2 , . . . , Prove that 〈a1, a2, . . . , an〉 įs an ideal of R. (If an ideal 1 = 〈a1, аг, . . . , an) for some a,...
B2. (a) Let I denote the interval 0,1 and let C denote the space of continuous functions I-R. Define dsup(f,g)-sup |f(t)-g(t) and di(f.g)f (t)- g(t)ldt (f,g E C) tEI (i) Prove that dsup is a metric on C (ii) Prove that di is a metric on C. (You may use any standard properties of continuous functions and integrals, provided you make your reasoning clear.) 6 i) Let 1 denote the constant function on I with value 1. Give an explicit...
(18) Let f and g be functions from R to R that have derivatives of al orders. Let h(k) denote the kth derivative of any function. Prove using the product rule for derivatives, the fact that and induction that k +1 k=0 (19) The Fibonacci numbers are defined recursively by Fn+2 = Fn+1 Prove that the number of subsets of { 1, 2, 3, . . . , n} containing no two successive integers is E, (20) Prove that 7n...
(6) Let A denote an m x n matrix. Prove that rank A < 1 if and only if A = BC. Where B is an m x 1 matrix and C is a 1 xn matrix. Solution (7) Do the following: (a) Use proof by induction to find a formula for for all positive integers n and for alld E R. Solution ... 2 for all positive (b) Find a closed formula for each entry of A" where A...
(4) Let R be an integral domain and o n,b,ce R such that ab = ne. Prove that b
4.5.17 Let V denote R^2 equipped with the l^ 1 norm and let W denote R^2 equipped with the l^∞ norm. Show that the linear map in L(V,W) represented by [ 1 1 ] is an isometry. Remark: On the other hand, if n ≥ 3, then R^n with the l^1 norm and R^n with the l^∞ norm are not isometric. [ 1 -1 ]