Mar 2006 Selected Problems, Chapter 3 Math 230(Mackey) Revised
... Since not all the α i ’s are zero, we have a non-trivial linear combination of all the vectors in S equalling the zero vector. This says that S is a linearly dependent set. But this contradicts what we were given! So our assumption that T is a linearly dependent set is false. Hence T is a linearly i ...
... Since not all the α i ’s are zero, we have a non-trivial linear combination of all the vectors in S equalling the zero vector. This says that S is a linearly dependent set. But this contradicts what we were given! So our assumption that T is a linearly dependent set is false. Hence T is a linearly i ...
Honors Algebra 4, MATH 371 Winter 2010
... M to be Tor(M ) := {m ∈ M : annR (m) 6= 0}. We say that an R-module N is torsion free if Tor(N ) = 0. Prove that Tor(M ) really is a submodule of M and that the quotient M/ Tor(M ) is torsion free. Solution: Suppose m, n ∈ Tor(M ) and that r, s ∈ R \ {0} are elements of annR (m) and annR (n), respec ...
... M to be Tor(M ) := {m ∈ M : annR (m) 6= 0}. We say that an R-module N is torsion free if Tor(N ) = 0. Prove that Tor(M ) really is a submodule of M and that the quotient M/ Tor(M ) is torsion free. Solution: Suppose m, n ∈ Tor(M ) and that r, s ∈ R \ {0} are elements of annR (m) and annR (n), respec ...
39(3)
... generalizations (see, e.g., [1], [3], [4], [5], [6], and [9] and the references therein). In this paper we shall try to treat some polynomial sequences by virtue of the line-sequential formalism developed earlier. To this end? we choose [9] as the guide of our endeavor and obtain some results of a d ...
... generalizations (see, e.g., [1], [3], [4], [5], [6], and [9] and the references therein). In this paper we shall try to treat some polynomial sequences by virtue of the line-sequential formalism developed earlier. To this end? we choose [9] as the guide of our endeavor and obtain some results of a d ...
(pdf)
... Now we consider again an arbitrary ring of integers of a finite extension, OK . It is clear that OK is a Z-module. However, it turns out that OK is actually a free Z-module of rank [K : Q]. The crucial step in showing this is to prove that OK is finitely generated over Z–the result then being immedi ...
... Now we consider again an arbitrary ring of integers of a finite extension, OK . It is clear that OK is a Z-module. However, it turns out that OK is actually a free Z-module of rank [K : Q]. The crucial step in showing this is to prove that OK is finitely generated over Z–the result then being immedi ...
Factorization
In mathematics, factorization (also factorisation in some forms of British English) or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 × 5, and the polynomial x2 − 4 factors as (x − 2)(x + 2). In all cases, a product of simpler objects is obtained.The aim of factoring is usually to reduce something to “basic building blocks”, such as numbers to prime numbers, or polynomials to irreducible polynomials. Factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. Viète's formulas relate the coefficients of a polynomial to its roots.The opposite of polynomial factorization is expansion, the multiplying together of polynomial factors to an “expanded” polynomial, written as just a sum of terms.Integer factorization for large integers appears to be a difficult problem. There is no known method to carry it out quickly. Its complexity is the basis of the assumed security of some public key cryptography algorithms, such as RSA.A matrix can also be factorized into a product of matrices of special types, for an application in which that form is convenient. One major example of this uses an orthogonal or unitary matrix, and a triangular matrix. There are different types: QR decomposition, LQ, QL, RQ, RZ.Another example is the factorization of a function as the composition of other functions having certain properties; for example, every function can be viewed as the composition of a surjective function with an injective function. This situation is generalized by factorization systems.