4. Suppose k is a field, A is a noetherian k-algebra and B is a finitely...
4) If A and B are mxn matrices and k and t are real numbers, prove that a)(A+B)k = Ak + Bk b) A(k + 1) = Ak + Al Note that to prove the result you cannot use specific examples. (HINT: Use the representative element notation for the matrices.)
Abstract Algebra based off of John B. Fraleigh's textbook 3. Find 473 (mod 15) 4. Find all integer solutions to the equation 21x 28 (mod 70). 5. Classify the group Z15 xZ4/K(3, 2)) using the fundamental theorem of finitely generated abelian groups.
part e and f 0 for all k E N and Σ at oo. For each of the following, either prove that the given series con- 4. Suppose ak verges, or provide an example for which the series diverges. ak 1 + at ar ai ak 0 for all k E N and Σ at oo. For each of the following, either prove that the given series con- 4. Suppose ak verges, or provide an example for which the series...
Please answer with the details. Thanks! In this problem using induction you prove that every finitely generated vector space has a basis. In fact, every vector space has a basis, but the proof of that is beyond the scope of this course Before trying this question, make sure you read the induction notes on Quercus. Let V be a non-zero initely generated vector space (1) Let u, Vi, . . . , v,e V. Prove tfe Span何, . . ....
Q7. QUESTION 5 5.1 (cf. Chapter 4, Exercise 129, p53) Let F be a field of characteristic not equal to 2 and let (K, a) be an P-algebra satislying the Jacobi identity, Show that K is a Lie algebra if and only if vu OK for all v e K. 5.2 Let F - GF(2). Define an operation on F by settingcShow that this some vEF operation turns Pt nto a Lie algebra such thatOp for so QUESTION 5 5.1...
Linear Algebra Suppose that B=P-1AP. (a) Prove that A and B have the same eigenvalues. (b) Prove that if x is an eigenvector of A, then P-10 is an edigenvector of B.
Part b.) 2. Let Bn be the ơ-algebra of all Borel sets in Rn and .Mn be the-algebra of all the measurable sets in Rn (a) Define Bn x Bk the a-algebra generated by "Borel rectangles" Bi x B2 with Bi E Bn and B2 E Bk. Prove that Bn x BB+k (b) Does a similar result hold for measurable sets, i.e. is MnXM-Mn+A? Here Mn x M is a σ.algebra generated by "Lebesgue rectangles" L1 ×し2 with Li E...
Problem 11.21. For k є Z, we define Ak-{x є Z : x-51+ k for some 1 є z} (a) Prove that {Ak : k Z} partitions Z. (b) We denote by ~ the equivalence relation on Z that is obtained from the par- tition of part (a). Give as simple a description ofas possible; that is, given condition "C(x,y)" on x and y s x~y if and only if "C(x, y)" holds. Problem 11.21. For k є Z, we...
(4) Let Σ ak and Σ bk be series with positive terms. The limit comparison test applies when a/bk L0; suppose for this problem that ak/bk0. (a) Show that if Σ bk converges, then Σ ak converges. Hint: remember we can delete finitely many terms from the series and not affect convergence. Use the fact a/bk0 to truncate the series at a convenient point. (b) Show that if ak diverges, then bk diverges. (c) Show by example that if Σ...