Notes in ring theory - University of Leeds
... (2.1.10) The power set P (S) of a set S is the set of all subsets. (2.1.11) A relation on S is an element of P (S × S). (2.1.12) A preorder is a reflexive transitive relation. Thus a poset is an antisymmetric preorder; and an equivalence is a symmetric preorder. An ordered set is a poset with every ...
... (2.1.10) The power set P (S) of a set S is the set of all subsets. (2.1.11) A relation on S is an element of P (S × S). (2.1.12) A preorder is a reflexive transitive relation. Thus a poset is an antisymmetric preorder; and an equivalence is a symmetric preorder. An ordered set is a poset with every ...
Solutions
... some root of unity λ. Solution: Since g has finite order, ρ(g)s = I for some s > 0. Hence, the minimal polynomial over of ρ(g) divides xs − 1, which has distinct roots in C. Thus, ρ(g) is diagonalizable. Let λ1 , · · · , λn be the eigenvalues of ρ(g). Then, as ρ(g)s = I, each λi is an sth root of un ...
... some root of unity λ. Solution: Since g has finite order, ρ(g)s = I for some s > 0. Hence, the minimal polynomial over of ρ(g) divides xs − 1, which has distinct roots in C. Thus, ρ(g) is diagonalizable. Let λ1 , · · · , λn be the eigenvalues of ρ(g). Then, as ρ(g)s = I, each λi is an sth root of un ...
Activity: Rational Exponents and Equations with Radicals
... the rational numbers: This set consists of all fractions of integers m/n, where n 6= 0. Two fractions a/b and m/n represent the same rational number if an = bm. For example, 2/3 and 10/15 represent the same rational number since 2(15) = 3(30). the real numbers: This consists of the sets of all lengt ...
... the rational numbers: This set consists of all fractions of integers m/n, where n 6= 0. Two fractions a/b and m/n represent the same rational number if an = bm. For example, 2/3 and 10/15 represent the same rational number since 2(15) = 3(30). the real numbers: This consists of the sets of all lengt ...
Mathematics Qualifying Exam University of British Columbia September 2, 2010
... 3. Let A be a square matrix with all diagonal entries equal to 2, all entries directly above or below the main diagonal equal to 1, and all other entries equal to 0. Show that every eigenvalue of A is a real number strictly between 0 and 4. ...
... 3. Let A be a square matrix with all diagonal entries equal to 2, all entries directly above or below the main diagonal equal to 1, and all other entries equal to 0. Show that every eigenvalue of A is a real number strictly between 0 and 4. ...
