The Mathematics 11 Competency Test
... by prime numbers for thousands of years, and continue to study their properties enthusiastically today because they have applications in many problems of technology. In simplifying fractions, it is useful to begin by factoring the numerator and denominator into a product of factors which are all pri ...
... by prime numbers for thousands of years, and continue to study their properties enthusiastically today because they have applications in many problems of technology. In simplifying fractions, it is useful to begin by factoring the numerator and denominator into a product of factors which are all pri ...
Matrix Analysis
... In other words, the eigenvectors of A associated with λ = 2 are all nonzero multiples of x = (4/5 1)T , and the eigenvectors associated with λ = 3 are all nonzero multiples of y = (1 1)T . Although there are an infinite number of eigenvectors associated with each eigenvalue, each eigenspace is one d ...
... In other words, the eigenvectors of A associated with λ = 2 are all nonzero multiples of x = (4/5 1)T , and the eigenvectors associated with λ = 3 are all nonzero multiples of y = (1 1)T . Although there are an infinite number of eigenvectors associated with each eigenvalue, each eigenspace is one d ...
Problem Seminar. Fall 2015. Problem Set 5. Combinatorics.
... that for each coloring, the sum of the numbers on the red squares is equal to the sum of the numbers on the black squares. ...
... that for each coloring, the sum of the numbers on the red squares is equal to the sum of the numbers on the black squares. ...
Full text
... Another approach would be to develop a q-bracket function (q-greatest integer function) and proceed in a manner similar to the above by expanding the binomial coefficient I . J in terms of a q-bracket function and using this in relation (2) just as we here used relation (2) in (16) to get (30) and t ...
... Another approach would be to develop a q-bracket function (q-greatest integer function) and proceed in a manner similar to the above by expanding the binomial coefficient I . J in terms of a q-bracket function and using this in relation (2) just as we here used relation (2) in (16) to get (30) and t ...
Problem Set 3
... Let ei be the element (guaranteed by the Chinese Remainder theorem) 1 (mod (t − µi )) and 0 (mod (t − µj )) for i 6= j, ei . Write these out explicitly (that is, as sums of elements of C3 . 3. Prove that a field of characteristic 3 cannot contain a cube root of unity. Let k be a field containing a c ...
... Let ei be the element (guaranteed by the Chinese Remainder theorem) 1 (mod (t − µi )) and 0 (mod (t − µj )) for i 6= j, ei . Write these out explicitly (that is, as sums of elements of C3 . 3. Prove that a field of characteristic 3 cannot contain a cube root of unity. Let k be a field containing a c ...
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.