Basic Concepts of Linear Algebra by Jim Carrell
... This textbook is meant to be an introduction to abstract linear algebra for first, second or third year university students who are specializing in mathematics or a closely related discipline. We hope that parts of this text will be relevant to students of computer science and the physical sciences. ...
... This textbook is meant to be an introduction to abstract linear algebra for first, second or third year university students who are specializing in mathematics or a closely related discipline. We hope that parts of this text will be relevant to students of computer science and the physical sciences. ...
CHAPTER 2: Linear codes
... Suppose a binary linear code is used only for error detection. The decoder will fail to detect errors which have occurred if the received word y is a codeword different from the codeword x which was sent, i. e. if the error vector e = y - x is itself a non-zero codeword. The probability Pundetect (C ...
... Suppose a binary linear code is used only for error detection. The decoder will fail to detect errors which have occurred if the received word y is a codeword different from the codeword x which was sent, i. e. if the error vector e = y - x is itself a non-zero codeword. The probability Pundetect (C ...
Algorithms for Matrix Canonical Forms
... for matrices over Z/(N ) but generalizes readily to matrices over an arbitrary principal ideal ring R. Howell’s proof of existence is constructive and leads to an O(n3 ) basic operations algorithm. When R is a field, the Howell form resolves to the reduced row echelon form and the Smith form to the ...
... for matrices over Z/(N ) but generalizes readily to matrices over an arbitrary principal ideal ring R. Howell’s proof of existence is constructive and leads to an O(n3 ) basic operations algorithm. When R is a field, the Howell form resolves to the reduced row echelon form and the Smith form to the ...
Linear Transformations
... Assume f : R2 → R2 is defined by f (x, y) = (x + y, x − y). Give the range of f and determine whether or not f is onto. Consider the functions in problems 6 and 7; one of them has the property that two distinctly different inputs are taken to the same output. This can be written (as an equation) as f ...
... Assume f : R2 → R2 is defined by f (x, y) = (x + y, x − y). Give the range of f and determine whether or not f is onto. Consider the functions in problems 6 and 7; one of them has the property that two distinctly different inputs are taken to the same output. This can be written (as an equation) as f ...
Linear Algebra II
... consisting of eigenvectors of f . The matrix representing f relative to this basis is then a diagonal matrix, with the various eigenvalues appearing on the diagonal. Since n × n matrices can be identified with endomorphisms F n → F n , all notions and results makes sense for square matrices, too. A ...
... consisting of eigenvectors of f . The matrix representing f relative to this basis is then a diagonal matrix, with the various eigenvalues appearing on the diagonal. Since n × n matrices can be identified with endomorphisms F n → F n , all notions and results makes sense for square matrices, too. A ...
Matrix Methods for Linear Systems of Differential Equations
... If we allow the entries a ij t in an n n matrix At to be functions of the variable t, then At is a matrix function of t. Similarly if the entries x i t of a vector xt are functions of t, then xt is a vector function of t. A matrix At is said to be continuous at t 0 if each a ij t i ...
... If we allow the entries a ij t in an n n matrix At to be functions of the variable t, then At is a matrix function of t. Similarly if the entries x i t of a vector xt are functions of t, then xt is a vector function of t. A matrix At is said to be continuous at t 0 if each a ij t i ...
Gröbner Bases of Bihomogeneous Ideals Generated - PolSys
... The complexity analysis that we perform by proving properties on the Hilbert biseries of bilinear ideals follows a path which is similar to the one used to analyze the complexity of the F5 algorithm in the case of homogeneous regular sequences (see Bardet et al. (2005)). In Kreuzer et al. (2002), th ...
... The complexity analysis that we perform by proving properties on the Hilbert biseries of bilinear ideals follows a path which is similar to the one used to analyze the complexity of the F5 algorithm in the case of homogeneous regular sequences (see Bardet et al. (2005)). In Kreuzer et al. (2002), th ...
Linear Algebra I
... consisting of eigenvectors of f . The matrix representing f relative to this basis is then a diagonal matrix, with the various eigenvalues appearing on the diagonal. Since n × n matrices can be identified with endomorphisms F n → F n , all notions and results makes sense for square matrices, too. A ...
... consisting of eigenvectors of f . The matrix representing f relative to this basis is then a diagonal matrix, with the various eigenvalues appearing on the diagonal. Since n × n matrices can be identified with endomorphisms F n → F n , all notions and results makes sense for square matrices, too. A ...
Solvable Groups, Free Divisors and Nonisolated
... In this first part of the paper, we identify a special class of representations of linear algebraic groups (especially solvable groups) which yield free divisors. Free divisors arising from representations are termed “linear free divisors”by Mond, who with Buchweitz first considered those that arise ...
... In this first part of the paper, we identify a special class of representations of linear algebraic groups (especially solvable groups) which yield free divisors. Free divisors arising from representations are termed “linear free divisors”by Mond, who with Buchweitz first considered those that arise ...
Fastest Mixing Markov Chain on Graphs with Symmetries
... The SDP formulation (2) means that the FMMC problem can be efficiently solved using standard SDP solvers, at least for small or medium size problems (with number of edges up to a thousand or so). General background on convex optimization and SDP can be found in, e.g., [NN94, VB96, WSV00, BTN01, BV04 ...
... The SDP formulation (2) means that the FMMC problem can be efficiently solved using standard SDP solvers, at least for small or medium size problems (with number of edges up to a thousand or so). General background on convex optimization and SDP can be found in, e.g., [NN94, VB96, WSV00, BTN01, BV04 ...
Determinants: Evaluation and Manipulation
... We will assume familiarity with basic properties of determinants. Just a reminder, if A = (aij )1≤i,j≤n is an n × n matrix, then X sgn(σ)a1σ(1) a2σ(2) · · · anσ(n) det A = σ∈Sn ...
... We will assume familiarity with basic properties of determinants. Just a reminder, if A = (aij )1≤i,j≤n is an n × n matrix, then X sgn(σ)a1σ(1) a2σ(2) · · · anσ(n) det A = σ∈Sn ...
MATLAB Exercises for Linear Algebra - M349 - UD Math
... If A and B are either scalars, vectors or matrices that can be added and if you type A+B in MATLAB you will get a correctly added sum. There is one way that MATLAB differs from standard mathematical usage of +. If A is any matrix, vector or scalar and b is any scalar then A + b is a matrix of the sa ...
... If A and B are either scalars, vectors or matrices that can be added and if you type A+B in MATLAB you will get a correctly added sum. There is one way that MATLAB differs from standard mathematical usage of +. If A is any matrix, vector or scalar and b is any scalar then A + b is a matrix of the sa ...
Spectral properties of the hierarchical product of graphs
... networks [4], and the neurons in the brain [5]. Large graphs and networks are often composed of several smaller pieces, for example motifs [6], communities or modules [7,8], layers [9], or self-similar subnetwork structures [10]. Moreover, the macroscopic properties of such large graphs are often de ...
... networks [4], and the neurons in the brain [5]. Large graphs and networks are often composed of several smaller pieces, for example motifs [6], communities or modules [7,8], layers [9], or self-similar subnetwork structures [10]. Moreover, the macroscopic properties of such large graphs are often de ...