Let G be a group, and let a ∈ G. Let φa : G −→ G be defined by φa(g) = aga−1 for all g ∈ G. (a) Prove that φa is an automorphism of G. (b) Let b ∈ G. What is the image of the element ba under the automorphism φa? (c) Why does this imply that |ab| = |ba| for all elements a, b ∈ G? 9. (5 points each) Let G be a group, and let...
Let be a metric space and let be the topology on induced by , and let be a compact space. Prove that is compact. (x, d) We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageAj,i=1,2,... na1 An
Let V = M2(R), and let U be the span of S = 2. (a) Let V = M,(R), and let U be the span of s={(1 1) ($ 3). (3), (1 9). (1) 2.)} Find a basis for U contained in S. (b) Let W be the subspace of P spanned by T = {2} + 22 – 1, -2.3 + 2x +1,23 +22² + 2x – 1, 2x3 + x2 +1 -2, 4.23 + 2x2 - -4}. Find...
Let I ⊆R be an open interval,let c ∈I,let f,g:I→R be functions and let k ∈R.Suppose that f and g are differentiable at c. f-g is differntiable at c and |f-g|' (c)=f'(c)-g'(c)
Let G be a finite group, and let H be a M be a subgroup of G such that H C M C G. What are the possible orders for M? Why? Let G possible orders of subgroups of S5 which contain D5? subgroup of G. Finally, let S5, and let H = D5. What are the _ Let G be a finite group, and let H be a M be a subgroup of G such that H C M...
Exercise 425 Let k and n be positive integers, let v eR”, and let A € Mkxn(R). Show that Av = 0 if and only if A? Av= 0.
Problem 9. Let ABCD be a parallelogram. Let E be a point on AD. Let F = BEN AC and G = BE CD. Prove BF = V FG* FE
Let R be a ring, let S be a subring of R and let' be an ideal of R. Note that I have proved that (5+1)/1 = {5 +1 | 5 € S) and I defined $:(5+1) ► S(SO ) by the formula: 0/5 + 1)=5+(SNI). In the previous video I showed that was well-defined. Now show that is a ring homomorphism. In other words, show that preserves both ring addition and ring multiplication. Then turn your work into this...
13.12.8 Problem. Let R be a ring and, let M be an R-module. Let m be a nonnegative integer, and suppose that M1,..., Mm are R-submodules of M, and that M is the internal direct sum of M1,..., Mm. Let n be a nonnegative integer with n < m, and for each i E {1,...,n}, let N; be an R-submodule of M. Let N = N1 ++ Nn. ... (i) Prove that N is the internal direct sum of N1,...,...
2. Let A = {Aq: a € A} be a family of sets and let B be a set. Prove that (a) Bn UA=U (BOA). αΕΔ QE A (b) Let 4 = {Aq: A E A} and let B = {Beß ef}. Use (a) to write (4) (Uda) (UB) UBR as a union of intersections.