Package `matrixcalc`
... Description A collection of functions to support matrix calculations for probability, econometric and numerical analysis. There are additional functions that are comparable to APL functions which are useful for actuarial models such as pension mathematics. This package is used for teaching and resea ...
... Description A collection of functions to support matrix calculations for probability, econometric and numerical analysis. There are additional functions that are comparable to APL functions which are useful for actuarial models such as pension mathematics. This package is used for teaching and resea ...
Algorithms for Matrix Canonical Forms
... each of these forms together with associated unimodular transformation matrices. The algorithms are analysed, with respect to the worst case, in terms of number of required operations from the ring. All algorithms are deterministic. For a square input matrix, the algorithms recover each of these for ...
... each of these forms together with associated unimodular transformation matrices. The algorithms are analysed, with respect to the worst case, in terms of number of required operations from the ring. All algorithms are deterministic. For a square input matrix, the algorithms recover each of these for ...
1: Introduction to Lattices
... the set L(B) is a lattice according to de nition De nition Notice that the lower bound mini kb∗i k depends on the choice of the basis. We will see later in the course that some bases give better lower bounds than others, but at this point any nonzero lower bound will su ce. We want to show that ther ...
... the set L(B) is a lattice according to de nition De nition Notice that the lower bound mini kb∗i k depends on the choice of the basis. We will see later in the course that some bases give better lower bounds than others, but at this point any nonzero lower bound will su ce. We want to show that ther ...
Problem set 9
... (b) Find the degree one polynomial p(x) which approximates sin(x) as well as possible R1 on [0, 1], in the sense that 0 (p(x) − sin(x))2 dx is as small as possible. (Suggestion: Project sin(x) onto the first two of the basis you found in Problem 9 last week. Use your answers from part (3a) and bilin ...
... (b) Find the degree one polynomial p(x) which approximates sin(x) as well as possible R1 on [0, 1], in the sense that 0 (p(x) − sin(x))2 dx is as small as possible. (Suggestion: Project sin(x) onto the first two of the basis you found in Problem 9 last week. Use your answers from part (3a) and bilin ...
linear old
... to problems beyond standard linear systems ones. Vectors in Rn and general vector spaces are presented back to back so that students are not stranded with the idea that vectors are just ordered lists of numbers. To this end, we also labor the notion of all functions from a set to the real numbers. I ...
... to problems beyond standard linear systems ones. Vectors in Rn and general vector spaces are presented back to back so that students are not stranded with the idea that vectors are just ordered lists of numbers. To this end, we also labor the notion of all functions from a set to the real numbers. I ...
Gröbner Bases of Bihomogeneous Ideals Generated - PolSys
... of the Macaulay matrices. Definition 5. Let Fi = (f1 , . . . , fi ) ∈ Ri be a sequence of homogeneous polynomials of degrees (d1 , . . . , di ) and ≺ be a monomial ordering. The Macaulay matrix in degree d Macaulay≺ (Fi , d) is the matrix whose rows contain the coefficients of the polynomials {tfj } ...
... of the Macaulay matrices. Definition 5. Let Fi = (f1 , . . . , fi ) ∈ Ri be a sequence of homogeneous polynomials of degrees (d1 , . . . , di ) and ≺ be a monomial ordering. The Macaulay matrix in degree d Macaulay≺ (Fi , d) is the matrix whose rows contain the coefficients of the polynomials {tfj } ...
Document
... the standard addition and the following nonstandard definition of scalar multiplication: c(x1, x2) = (cx1, 0). Show that V is not a vector space. pf: This example satisfies the first nine axioms of the definition of a vector space. For example, let u = (1, 1), v = (3, 4), and c = 2, then we have c(u ...
... the standard addition and the following nonstandard definition of scalar multiplication: c(x1, x2) = (cx1, 0). Show that V is not a vector space. pf: This example satisfies the first nine axioms of the definition of a vector space. For example, let u = (1, 1), v = (3, 4), and c = 2, then we have c(u ...
Linearly independence Definition: Consider a set of n
... set of vectors spans V 3 H R L . In other words, given any arbitrary vector H = @ a , b , c D T , it can always be expressed as a linear combanation based on the vectors {K 1 ,K 2 ,K 3 , K 4 }. Note: The exact values of x 1 , x 2 , x 3 , x 4 are not important here. What is important is that the ...
... set of vectors spans V 3 H R L . In other words, given any arbitrary vector H = @ a , b , c D T , it can always be expressed as a linear combanation based on the vectors {K 1 ,K 2 ,K 3 , K 4 }. Note: The exact values of x 1 , x 2 , x 3 , x 4 are not important here. What is important is that the ...
3. Linear function
... Since the first components (in red) do not match up, we suspect that F is not linear. We cannot write F (x + y) 6= F (x) + F (y), though, since there are choices for x and y that actually give equality (for instance, x = 0 and y = 0). However, in order to show that F fails property (a) it is enough ...
... Since the first components (in red) do not match up, we suspect that F is not linear. We cannot write F (x + y) 6= F (x) + F (y), though, since there are choices for x and y that actually give equality (for instance, x = 0 and y = 0). However, in order to show that F fails property (a) it is enough ...
Fundamentals of Linear Algebra
... supplement of standard textbooks or as a textbook for a formal course in linear algebra. This book is not a ”traditional” book in the sense that it does not include any applications to the material discussed. Its aim is solely to learn the basic theory of linear algebra within a semester period. Ins ...
... supplement of standard textbooks or as a textbook for a formal course in linear algebra. This book is not a ”traditional” book in the sense that it does not include any applications to the material discussed. Its aim is solely to learn the basic theory of linear algebra within a semester period. Ins ...
CHAPTER 2: Linear codes
... IV054 Linear codes Linear codes are special sets of words of the length n over an alphabet {0,..,q -1}, where q is a power of prime. Since now on sets of words Fqn will be considered as vector spaces V(n,q) of vectors of length n with elements from the set {0,..,q -1} and arithmetical operations wi ...
... IV054 Linear codes Linear codes are special sets of words of the length n over an alphabet {0,..,q -1}, where q is a power of prime. Since now on sets of words Fqn will be considered as vector spaces V(n,q) of vectors of length n with elements from the set {0,..,q -1} and arithmetical operations wi ...
Learning mixtures of product distributions over
... Such learning problems have been well studied in the past, as we now describe. 1.2 Related work. In [18] Kearns et al. gave efficient algorithms for learning mixtures of Hamming balls; these are product distributions over {0, 1}n in which all the coordinate means E[Xij ] must be either p or 1 − p fo ...
... Such learning problems have been well studied in the past, as we now describe. 1.2 Related work. In [18] Kearns et al. gave efficient algorithms for learning mixtures of Hamming balls; these are product distributions over {0, 1}n in which all the coordinate means E[Xij ] must be either p or 1 − p fo ...
