the fundamental theorem of linear algebra
Tài liệu Solution of Linear Algebraic Equations part 1 docx
... zero elements except for in the jth component) The corresponding x’s are then the columns of the matrix inverse of A (§2.1 and §2.3) • Calculation of the determinant of a square matrix A (§2.3) ... Introduction Much of the sophistication of complicated linear equation-solving packages” is devoted to the detection and/or correction of these two pathologies As you work with large linear sets of equations, ... event, the solution space consists of a particular solution xp added to any linear combination of (typically) N − M vectors (which are said to be in the nullspace of the matrix A) The task of finding...
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Tài liệu Solution of Linear Algebraic Equations part 2 ppt
... row, as long as we the same linear combination of the rows of the b’s and (which then is no longer the identity matrix, of course) • Interchanging any two columns of A gives the same solution ... corresponding rows of the x’s and of Y In other words, this interchange scrambles the order of the rows in the solution If we this, we will need to unscramble the solution by restoring the rows to their original ... other row) Then the right amount of the first row is subtracted from each other row to make all the remaining ai1 ’s zero The first column of A now agrees with the identity matrix We move to the...
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Tài liệu Solution of Linear Algebraic Equations part 11 ppt
... Constructs the QR decomposition of a[1 n][1 n] The upper triangular matrix R is returned in the upper triangle of a, except for the diagonal elements of R which are returned in d[1 n] The orthogonal ... float b[]) Solves the set of n linear equations A · x = b a[1 n][1 n], c[1 n], and d[1 n] are input as the output of the routine qrdcmp and are not modified b[1 n] is input as the right-hand side ... elements in a column of the matrix situated below a chosen element Thus we arrange for the first Householder matrix Q1 to zero all elements in the first column of A below the first element Similarly...
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Tài liệu Solution of Linear Algebraic Equations part 3 pdf
... computing the inverse matrix as part of the Gauss-Jordan scheme.) For computing the inverse matrix (which we can view as the case of M = N right-hand sides, namely the N unit vectors which are the ... the linear set A · x = (L · U) · x = L · (U · x) = b (2.3.3) by first solving for the vector y such that L·y=b (2.3.4) U·x=y (2.3.5) and then solving What is the advantage of breaking up one linear ... the matrix is reduced, and the increasing numbers of predictable zeros reduce the count to one-third), and N M times, respectively Each backsubstitution of a right-hand side is N executions of...
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Tài liệu Solution of Linear Algebraic Equations part 4 docx
... can modify the loop of the above fragment and (e.g.) divide by powers of ten, to keep track of the scale separately, or (e.g.) accumulate the sum of logarithms of the absolute values of the factors ... Go back for the next column in the reduction free_vector(vv,1,n); 48 Chapter Solution of Linear Algebraic Equations To summarize, this is the preferred way to solve the linear set of equations ... accurate The determinant of an LU decomposed matrix is just the product of the diagonal elements, N det = βjj (2.3.15) j=1 We don’t, recall, compute the decomposition of the original matrix, but rather...
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Tài liệu Solution of Linear Algebraic Equations part 5 docx
... possible within the storage limitations of bandec, and the above routine does take advantage of the opportunity In general, when TINY is returned as a diagonal element of U , then the original matrix ... diagonal elements of U (whose product, times d = ±1, gives the determinant) are returned in the first column of A’s storage space The following routine, bandec, is the band-diagonal analog of ludcmp in ... Linear Algebraic Equations In that case, the solution of the linear system by LU decomposition can be accomplished much faster, and in much less storage, than for the general N × N case The precise...
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Tài liệu Solution of Linear Algebraic Equations part 12 pdf
... (2.11.3) the a’s and b’s are never commuted Therefore (2.11.3) and (2.11.4) are valid when the a’s and b’s are themselves matrices The problem of multiplying two very large matrices (of order ... it reduces the process of matrix multiplication to order N log2 instead of N What about all the extra additions in (2.11.3)–(2.11.4)? Don’t they outweigh the advantage of the fewer multiplications? ... matrix inversion [1] Suppose that the matrices a11 a21 a12 a22 and c11 c21 c12 c22 (2.11.5) are inverses of each other Then the c’s can be obtained from the a’s by the following operations (compare...
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Tài liệu Solution of Linear Algebraic Equations part 6 pptx
... approximately the identity matrix Define the residual matrix R of B0 as 58 Chapter Solution of Linear Algebraic Equations We can define the norm of a matrix as the largest amplification of length that ... on the following theorem of linear algebra, whose proof is beyond our scope: Any M × N matrix A whose number of rows M is greater than or equal to its number of columns N , can be written as the ... x[1 n] of the linear set of equations A · X = B The matrix a[1 n][1 n], and the vectors b[1 n] and x[1 n] are input, as is the dimension n Also input is alud[1 n][1 n], the LU decomposition of a...
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Tài liệu Solution of Linear Algebraic Equations part 7 docx
... called the nullity of A Now, there is also some subspace of b that can be “reached” by A, in the sense that there exists some x which is mapped there This subspace of b is called the range of A The ... diagnosis of the situation Formally, the condition number of a matrix is defined as the ratio of the largest (in magnitude) of the wj ’s to the smallest of the wj ’s A matrix is singular if its ... vector space b If A is singular, then there is some subspace of x, called the nullspace, that is mapped to zero, A · x = The dimension of the nullspace (the number of linearly independent vectors...
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Tài liệu Solution of Linear Algebraic Equations part 8 docx
... that contains the first off-diagonal element of the corresponding row of the matrix (If there are no off-diagonal elements for that row, it is one greater than the index in sa of the most recently ... vector ei , then (2.7.1) adds the components of v to the ith row (Recall that u ⊗ v is a matrix whose i, jth element is the product of the ith component of u and the jth component of v.) If v ... correction of the form u ⊗ v, and solve the linear system 2.7 Sparse Linear Systems 75 Here γ is arbitrary for the moment Then the matrix A is the tridiagonal part of the matrix in (2.7.9), with two...
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Tài liệu Solution of Linear Algebraic Equations part 9 docx
... use the fact that the inverse of the transpose is the transpose of the inverse, so N cj = Akj yk (2.8.6) k=1 The routine in §3.5 implements this It remains to find a good way of multiplying out the ... be the polynomial of degree N − defined by N Pj (x) = n=1 (n=j) N x − xn = Ajk xk−1 xj − xn k=1 (2.8.3) Here the meaning of the last equality is to define the components of the matrix Aij as the ... will see that it relates to the problem of moments: Given the values of N points xi , find the unknown weights wi , assigned so as to match the given values qj of the first N moments (For more...
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Tài liệu Solution of Linear Algebraic Equations part 10 docx
... use the fact that the inverse of the transpose is the transpose of the inverse, so N cj = Akj yk (2.8.6) k=1 The routine in §3.5 implements this It remains to find a good way of multiplying out the ... the case of a tridiagonal matrix was treated specially, because that particular type of linear system admits a solution in only of order N operations, rather than of order N for the general linear ... with the fitting of polynomials, the reconstruction of distributions from their moments, and also other contexts In this book, for example, a Vandermonde problem crops up in §3.5 Matrices of the...
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Tài liệu The Fundamental Principles of Old and New World Civilizations docx
... Campeche The extremity of the wing of the land is Calkini; the (chun) place where the wing grows or begins, is Izamal The half of the wing is Zaci; the tip of the wing is Cumkal The head of the land ... representation of the face of "the lord of the North," in fig 19, gives the impression that it was also used to convey the idea of duality, or the union of The Fundamental Principles of Old and New ... and to offer it to "the Sun whom they called father and to the earth their mother." They severed its head and raised this as though offering it to the sun They then tilled the earth where the blood...
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THE FUNDAMENTAL ROLE OF SCIENCE AND TECHNOLOGYIN INTERNATIONAL DEVELOPMENT pptx
... secure the services of eminent members of appropriate professions in the examination of policy matters pertaining to the health of the public The Institute acts under the responsibility given to the ... and the 1970s saw tremendous swings from the latter to the former, setting the stage for another swing of the pendulum back to the latter as a result of disappointments with the results of diffuse ... members of the committee on the field visits and who provided general guidance to the committee greatly enriched the quality of the report: Michael Clegg, Foreign Secretary of the National Academy of...
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handbook of linear algebra
... (associative) algebra Boundary The boundary ∂S of a subset S of the real numbers or the complex numbers is the intersection of the closure of S and the closure of the complement of S Examples: The boundary ... numerical linear algebra, two important branches of the subject Applications of linear algebra to other disciplines, both inside and outside of mathematics, comprise the fourth part of the book ... linear algebra are not redefined in each chapter The Glossary, covering the terminology of linear algebra, combinatorial linear algebra, and numerical linear algebra, is available at the end of the...
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Báo cáo toán học: "The Fundamental Group of Balanced Simplicial Complexes and Posets" potx
... Thus the Hurewicz Theorem says that the minimal number of generators of the fundamental group of a simplicial complex ∆ is greater than or equal to the number of generators of H1 (∆; Z) By the ... a description of the fundamental group in terms of a finite set of generators and relations In Section 4, we use the theorems in Section to prove Theorem 4.5 This theorem gives the desired bound ... group of ∆ based at v0 The Cellular Approximation Theorem ([10] VII.6.17) tells us that any path in ∆ is homotopic to a path in the 1-skeleton of ∆ We use this fact to motivate the proof of the...
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on the fundamental group of noncompact manifolds with nonnegative ricci curvature
... and 3.1.9 First we present the proof of Theorem 3.1.8 Proof of Theorem 3.1.8 Fix ε0 > Let Γr be the set of halfway generators Γr By the Halfway Lemma, if there exists L of G(p, r, (1+ε0 )r) and ... Proof of Theorem 3.1.5 In this section we present the proof of Theorem 3.1.5 The general idea is to consider a loop γ that satisfies the hypotheses of Lemma 3.4.4 Let σ be a ray at p Then, by the triangle ... important fact in the proof of Theorems 3.1.5 and 3.1.8 To end the section we prove Lemma 3.2.9 which we will apply in the proofs of Theorems 3.1.5 and 3.1.8 To motivate the definition of nullhomotopy...
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homomorphisms of the fundamental group of a surface into psu(1,1), and the action of the mapping class group
... is going to be crucial for the proof of the ergodicity of the mapping class group In section 6.3 we state the main theorem of this paper To proceed the proof of the theorem we state and prove ... help us use some of Goldman’s results to complete the proof of our theorem In chapter 7, we discuss the sewing lemma (the analogue for homomorphisms of the Seifert-Van Kampen theorem) Our original ... conj denote the projection The idea of the proof is to prove first that the space of conjugacy classes of Im(Rp−1 ) ∩ gc Im(R1 ) is connected, then use the fact that each ˜ one of the conjugacy...
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