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'Secure Computation' has been a classic and central question in modern cryptography with a large set of potential applications.Mining large genomic databases, private scientific computation, and studying properties of shared networks are just a few examples.Unfortunately, the majority of the constructions in this area have not made their way into practice, primarily due to their inefficiency. In this talk, I first outline three different approaches toward designing more practical protocols, and briefly describe some of our results in each direction. I will then focus on one approach and the problem of 'Secure Linear Algebra'. In a distributed system, linear constraints might reveal sensitive information and therefore, the involved parties want to reveal as little information as possible about their private inputs. I consider this problem in multiple distributed settings and adversarial environments. In each case, I will describe protocols that are nearly optimal in terms of round, computation, and communication efficiency. ©2009 Microsoft Corporation. All rights reserved.

Linear Algebra and Its Applications 2004: Volume 381 , Issue Index. Digitized from IA1652402-05 . Previous issue: sim_linear-algebra-and-its-applications_2004-03-15_380 . Next issue: sim_linear-algebra-and-its-applications_2004_381_contents .

Linear algebra course by Andronick Arutyunov at Free Moscow University, fall 2021, Moscow tutorials by Alex Shpilkin, tutorial 9 on 2021-12-12.

Linear Algebra and Its Applications 1987: Volume 92 , Issue Index. Digitized from IA1653332-04 . Previous issue: sim_linear-algebra-and-its-applications_1987-06_91_0 . Next issue: sim_linear-algebra-and-its-applications_1987_92_contents .

Linear Algebra and Its Applications 1994: Volume 199 , Issue Index. Digitized from IA1653332-03 . Previous issue: sim_linear-algebra-and-its-applications_january-15-february-1994_197-198_0 . Next issue: sim_linear-algebra-and-its-applications_1994_199_contents_0 .

Aula 9 (1.10.21) - Produtos escalares: motivação (produto escalar canônico em Rn), definição e axiomas (simetria, positividade definida e bilinearidade). O produto escalar associado a uma base finita. Ortogonalidade, conjuntos ortogonais e ortonormais, normalização de conjuntos ortogonais que não contém zero. Bases ortogonais e ortonormais, normalização de uma base ortogonal, unicidade do produto escalar associado a uma base finita, componentes de um vetor em bases ortogonais e ortonormais.

In the paper I considered linear and antilinear automorphisms of quaternion algebra. I proved the theorem that there is unique expansion of R-linear mapping of quaternion algebra relative to the given set of linear and antilinear automorphisms.

Linear Algebra and Its Applications 2001: Volume 324 , Issue Index. Digitized from IA1653332-07 . Previous issue: sim_linear-algebra-and-its-applications_2001-01-15_323 . Next issue: sim_linear-algebra-and-its-applications_2001_324_contents_0 .

Linear Algebra and Its Applications 1990: Volume 128 , Issue Contents. Digitized from IA1653332-01 . Previous issue: sim_linear-algebra-and-its-applications_1990_128_index_0 . Next issue: sim_linear-algebra-and-its-applications_1990-01_128_0 .

Aula 19 (14.4.22) - Transformações lineares (vi): pseudoinversas de transformações lineares - definição, projeções lineares associadas a uma pseudoinversa. Interlúdio: propriedades de projeções lineares. O caso de pseudoinversas de uma transformação linear que são "pseudoinvertidas" pela última: determinação pelas projeções lineares associadas. O caso de projeções ortogonais: a pseudoinversa de Moore-Penrose, propriedade de minimização da distância à imagem.

Aula 19 (14.4.22) - Transformações lineares (vi): pseudoinversas de transformações lineares - definição, projeções lineares associadas a uma pseudoinversa. Interlúdio: propriedades de projeções lineares. O caso de pseudoinversas de uma transformação linear que são "pseudoinvertidas" pela última: determinação pelas projeções lineares associadas. O caso de projeções ortogonais: a pseudoinversa de Moore-Penrose, propriedade de minimização da distância à imagem.

Linear Algebra and Its Applications 1999: Volume 288 , Issue CONTENTS. Digitized from IA1652402-06 . Previous issue: sim_linear-algebra-and-its-applications_1999_288_index . Next issue: sim_linear-algebra-and-its-applications_1999-02-01_288 .

Linear Algebra and Its Applications 1997: Volume 262 , Issue CONTENTS. Digitized from IA1653332-06 . Previous issue: sim_linear-algebra-and-its-applications_1997_262_index_0 . Next issue: sim_linear-algebra-and-its-applications_1997-09-01_262_0 .

Linear Algebra and Its Applications 1997: Volume 262 , Issue CONTENTS. Digitized from IA1653332-06 . Previous issue: sim_linear-algebra-and-its-applications_1997_262_index_0 . Next issue: sim_linear-algebra-and-its-applications_1997-09-01_262_0 .

Linear Algebra and Its Applications 1992: Volume 168 , Issue CONTENTS. Digitized from IA1652402-02 . Previous issue: sim_linear-algebra-and-its-applications_1992_168_index . Next issue: sim_linear-algebra-and-its-applications_1992-04-15_168 .

Linear Algebra and Its Applications 1992: Volume 168 , Issue CONTENTS. Digitized from IA1652402-02 . Previous issue: sim_linear-algebra-and-its-applications_1992_168_index . Next issue: sim_linear-algebra-and-its-applications_1992-04-15_168 .

Linear Algebra and Its Applications 2004: Volume 381 , Issue CONTENTS. Digitized from IA1653333-02 . Previous issue: sim_linear-algebra-and-its-applications_2004_381_index_0 . Next issue: sim_linear-algebra-and-its-applications_2004-04-01_381_0 .

Linear Algebra and Its Applications 1987: Volume 96 , Issue Contents. Digitized from IA1653332-04 . Previous issue: sim_linear-algebra-and-its-applications_1987_96_index . Next issue: sim_linear-algebra-and-its-applications_1987-11_96_0 .

Certificates to a linear algebra computation are additional data structures for each output, which can be used by a---possibly randomized---verification algorithm that proves the correctness of each output. The certificates are essentially optimal if the time (and space) complexity of verification is essentially linear in the input size $N$, meaning $N$ times a factor $N^{o(1)}$, i.e., a factor $N^{\eta(N)}$ with $\lim_{N\to \infty} \eta(N)$ $=$ $0$. We give algorithms that compute essentially optimal certificates for the positive semidefiniteness, Frobenius form, characteristic and minimal polynomial of an $n\times n$ dense integer matrix $A$. Our certificates can be verified in Monte-Carlo bit complexity $(n^2 \lognormA)^{1+o(1)}$, where $\lognormA$ is the bit size of the integer entries, solving an open problem in [Kaltofen, Nehring, Saunders, Proc.\ ISSAC 2011] subject to computational hardness assumptions. Second, we give algorithms that compute certificates for the rank of sparse or structured $n\times n$ matrices over an abstract field, whose Monte Carlo verification complexity is $2$ matrix-times-vector products $+$ $n^{1+o(1)}$ arithmetic operations in the field. For example, if the $n\times n$ input matrix is sparse with $n^{1+o(1)}$ non-zero entries, our rank certificate can be verified in $n^{1+o(1)}$ field operations. This extends also to integer matrices with only an extra $||A||^{1+o(1)}$ factor. All our certificates are based on interactive verification protocols with the interaction removed by a Fiat-Shamir identification heuristic. The validity of our verification procedure is subject to standard computational hardness assumptions from cryptography.

Linear Algebra and Its Applications 1988: Volume 105 , Issue CONTENTS. Digitized from IA1652402-02 . Previous issue: sim_linear-algebra-and-its-applications_1988_105_index_0 . Next issue: sim_linear-algebra-and-its-applications_1988-07_105_0 .

Linear Algebra and Its Applications 1998: Volume 285 , Issue CONTENTS. Digitized from IA1653332-05 . Previous issue: sim_linear-algebra-and-its-applications_1998_285_index . Next issue: sim_linear-algebra-and-its-applications_1998-12-01_285 .

Aula 24 (28.4.22) - Sistemas lineares (iii): recapitulação - descrição do algoritmo de eliminação gaussiana ( = escalonamento de matrizes). Exemplos de escalonamento de sistemas lineares por eliminação gaussiana: (1.) sistema com uma única solução, substituição de variáveis nos pivôs da matriz escalonada; (2.) sistema com mais de uma solução; (3.) sistema sem solução, critério de existência de soluções em termos do sistema escalonado.

This guide presents lesson plans for the teaching of mathematics within the biomedical curriculum. It is designed to accompany the student text. Lessons concentrate on biomedical applications of the subject mathematical techniques, and are designed to be taught in close cooperation with the science lessons of the curriculum. Each lesson plan presents objectives, class periods required, overview and remarks, and a key to problems presented in the student text. (RE)

YouTube: Curso de Álgebra Linear

Linear Algebra and Its Applications 1987: Volume 90 , Issue Contents. Digitized from IA1653332-04 . Previous issue: sim_linear-algebra-and-its-applications_1987_90_index . Next issue: sim_linear-algebra-and-its-applications_1987-05_90_0 .

We consider the relations of generalized commutativity in the algebra of formal series $ M_q (x^i ) $, which conserve a tensor $ I_q $-grading and depend on parameters $ q(i,k) $ . We choose the $ I_q $-preserving version of differential calculus on $ M_q$ . A new construction of the symmetrized tensor product for $ M_q $-type algebras and the corresponding definition of minimally deformed linear group $ QGL(n) $ and Lie algebra $ qgl(n) $ are proposed. We study the connection of $ QGL(n) $ and $ qgl(n) $ with the special matrix algebra $ \mbox{Mat} (n,Q) $ containing matrices with noncommutative elements. A definition of the deformed determinant in the algebra $ \mbox{Mat} (n,Q) $ is given. The exponential parametrization in the algebra $ \mbox{Mat} (n,Q) $ is considered on the basis of Campbell-Hausdorf formula.

Linear Algebra and Its Applications 2001: Volume 323 , Issue CONTENTS. Digitized from IA1652402-05 . Previous issue: sim_linear-algebra-and-its-applications_2001_323_index . Next issue: sim_linear-algebra-and-its-applications_2001-01-15_323_1-3 .

The space of linear differential operators on a smooth manifold $M$ has a natural one-parameter family of $Diff(M)$ (and $Vect(M)$)-module structures, defined by their action on the space of tensor-densities. It is shown that, in the case of second order differential operators, the $Vect(M)$-module structures are equivalent for any degree of tensor-densities except for three critical values: $\{0,{1\over 2},1\}$. Second order analogue of the Lie derivative appears as an intertwining operator between the spaces of second order differential operators on tensor-densities.

calculo 1 

In the present chapter, we focus our attention on eigenvalues and eigenvectors, which is probably the most important characterization of a linear operator that is available to us. In Chapter 8 we give a complete discussion of canonical forms of matrices under similarity

Linear Algebra and Its Applications 2005: Volume 410 , Issue CONTENTS. Digitized from IA1653333-02 . Previous issue: sim_linear-algebra-and-its-applications_2005-11-01_409_0 . Next issue: sim_linear-algebra-and-its-applications_2005_410_index .

Linear Algebra and Its Applications 1993: Volume 181 , Issue Index. Digitized from IA1652402-02 . Previous issue: sim_linear-algebra-and-its-applications_1993-02_180 . Next issue: sim_linear-algebra-and-its-applications_1993_181_contents .

Proof by induction that transposing a matrix does not change its determinant

Linear Algebra and Its Applications 1993: Volume 182 , Issue Index. Digitized from IA1653332-02 . Previous issue: sim_linear-algebra-and-its-applications_1993-03-01_181_0 . Next issue: sim_linear-algebra-and-its-applications_1993_182_contents_0 .

Isomorphisms of separable Hilbert spaces are analogous to isomorphisms of n-dimensional vector spaces. However, while n-dimensional spaces in applications are always realized as the Euclidean space R^n, Hilbert spaces admit various useful realizations as spaces of functions. In the paper this simple observation is used to construct a fruitful formalism of local coordinates on Hilbert manifolds. Images of charts on manifolds in the formalism are allowed to belong to arbitrary Hilbert spaces of functions including spaces of generalized functions. Tensor equations then describe families of functional equations on various spaces of functions. The formalism itself and its applications in linear algebra, differential equations and differential geometry are analyzed.

A Linear Associative Algebra Suitable for Electromagnetic Relations and the Theory of Relativity. Johnston, W. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character (1905-1934). 1919-12-15. 96 (678):331–333

Aula 20 (3.8.22) - Transformações lineares (v): recapitulação - determinação de uma transformação linear por seus valores numa base, vetor-coluna ( = matriz com uma única coluna) de um vetor numa base, forma matricial da ação de transformações lineares num par de bases, a matriz de uma transformação linear. Forma matricial das operações algébricas sobre transformações lineares: operações vetoriais pontuais, produto e inversa. Transposição de matrizes e a adjunta de uma transformação linear - identidade fundamental. O efeito de mudança de bases sobre a matriz de uma transformação linear - enunciado (páginas 43 a 48 das notas de aula).

We present here algorithms for efficient computation of linear algebra problems over finite fields.

It is useful for CSIR NET Mathematics.

Aula 13 (24.3.22) - Produtos escalares (iii): projeções ortogonais. Expansão e componentes de um vetor num (sub)espaço vetorial numa base ortonormal ou apenas ortogonal. Extensão da expansão de vetores num subespaço vetorial numa base ortogonal ao espaço ambiente = definição de projeção ortogonal ao longo do subespaço vetorial. Propriedades (linearidade, idempotência, imagem e ortogonalidade). Decomposição ortogonal de vetores ao longo de um subespaço vetorial, independência de projeções ortogonais com respeito à escolha de base. Caracterização geométrica da projeção ortogonal de um vetor ao longo de um subespaço vetorial como o vetor no subespaço mais próximo do vetor dado. Ortogonalização e ortonormalização de bases usando projeções ortogonais: o algoritmo de Gram-Schmidt.

Linear Algebra and Its Applications 2000: Volume 317 , Issue Contents. Digitized from IA1653333-03 . Previous issue: sim_linear-algebra-and-its-applications_2000_317_index_0 . Next issue: sim_linear-algebra-and-its-applications_2000-09-15_317_1-3_0 .

Linear Algebra and Its Applications 2000: Volume 317 , Issue Contents. Digitized from IA1653333-03 . Previous issue: sim_linear-algebra-and-its-applications_2000_317_index_0 . Next issue: sim_linear-algebra-and-its-applications_2000-09-15_317_1-3_0 .

The Cayley--Hamilton--Newton theorem for half-quantum matrices is proven.

Using the differential precision methods developed previously by the same authors, we study the p-adic stability of standard operations on matrices and vector spaces. We demonstrate that lattice-based methods surpass naive methods in many applications, such as matrix multiplication and sums and intersections of subspaces. We also analyze determinants , characteristic polynomials and LU factorization using these differential methods. We supplement our observations with numerical experiments.

Linear Algebra and Its Applications 1988: Volume 109 , Issue Index. Digitized from IA1653331-06 . Previous issue: sim_linear-algebra-and-its-applications_1988-09_108 . Next issue: sim_linear-algebra-and-its-applications_1988_109_contents .

Linear Algebra and Its Applications 1990: Volume 139 , Issue Index. Digitized from IA1652402-02 . Previous issue: sim_linear-algebra-and-its-applications_august-september-1990_137-138 . Next issue: sim_linear-algebra-and-its-applications_1990_139_contents .

Linear Algebra and Its Applications 1993: Volume 186 , Issue Index. Digitized from IA1652402-02 . Previous issue: sim_linear-algebra-and-its-applications_1993-05_185 . Next issue: sim_linear-algebra-and-its-applications_1993_186_contents .

Linear Algebra and Its Applications 1993: Volume 191 , Issue CONTENTS. Digitized from IA1652402-02 . Previous issue: sim_linear-algebra-and-its-applications_1993_191_index . Next issue: sim_linear-algebra-and-its-applications_1993-09-15_191 .

Linear Algebra and Its Applications 1993: Volume 192 , Issue CONTENTS. Digitized from IA1652402-02 . Previous issue: sim_linear-algebra-and-its-applications_1993_192_index . Next issue: sim_linear-algebra-and-its-applications_1993-10_192 .

Linear Algebra and Its Applications 1995: Volume 214 , Issue Index. Digitized from IA1652402-06 . Previous issue: sim_linear-algebra-and-its-applications_1988-1994_101-200_index . Next issue: sim_linear-algebra-and-its-applications_1995_214_contents .