Topic
Coordinate vector
About: Coordinate vector is a research topic. Over the lifetime, 388 publications have been published within this topic receiving 7686 citations.
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Papers
Book•
1 May 2000
TL;DR: In this paper, the authors present an overview of the history of linear algebra and its applications in computer graphics and computer networks, including the following: 1.1 Introduction to Systems of Linear Equations and Matrices. 2.1 Determinants by Cofactor Expansion. 3.2 Evaluating determinants by Row Reduction. 4.3 Properties of the Determinant Function.
Abstract: Chapter 1. Systems of Linear Equations and Matrices. 1.1 Introduction to Systems of Linear Equations. 1.2 Gaussian Elimination. 1.3 Matrices and Matrix Operations. 1.4 Inverses Rules of Matrix Arithmetic. 1.5 Elementary Matrices and a Method for Finding A-1. 1.6 Further Results on Systems of Equations and Invertibility. 1.7 Diagonal, Triangular, and Symmetric Matrices. Chapter 2. Determinants. 2.1 Determinants by Cofactor Expansion. 2.2 Evaluating Determinants by Row Reduction. 2.3 Properties of the Determinant Function. 2.4 A Combinatorial Approach to Determinants. Chapter 3. Vectors in 2 Space and 3-Space. 3.1 Introduction to Vectors (Geometric). 3.2 Norm of a Vector Vector Arithmetic. 3.3 Dot Product Projections. 3.4 Cross Product. 3.5 Lines and Planes in 3-Space. Chapter 4. Euclidean Vector Spaces. 4.1 Euclidean n-Space. 4.2 Linear Transformations from Rn to Rm. 4.3 Properties of Linear Transformations from Rn to Rm. 4.4 Linear Transformations and Polynomials. Chapter 5. General Vector Spaces. 5.1 Real Vector Spaces. 5.2 Subspaces. 5.3 Linear Independence. 5.4 Basis and Dimension. 5.5 Row Space, Column Space, and Nullspace. 5.6 Rank and Nullity. Chapter 6. Inner Product Spaces. 6.1 Inner Products. 6.2 Angle and Orthogonality in Inner Product Spaces. 6.3 Orthonormal Bases: Gram-Schmidt Prodcess QR-Decomposition. 6.4 Best Approximation Least Squares. 6.5 Change of Basis. 6.6 Orthogonal Matrices. Chapter 7. Eigenvalues, Eigenvectors. 7.1 Eigenvalues and Eigenvectors. 7.2 Diagonalization. 7.3 Orthogonal Diagonalization. Chapter 8. Linear Transformations. 8.1 General Linear Transformations. 8.2 Kernel and range. 8.3 Inverse Linear Transformations. 8.4 Matrices of General Linear Transformations. 8.5 Similarity. 8.6 Isomorphism. Chapter 9. Additional topics. 9.1 Application to Differential Equations. 9.2 Geometry and Linear Operators on R2. 9.3 Least Squares Fitting to Data. 9.4 Approximation Problems Fourier Series. 9.5 Quadratic Forms. 9.6 Diagonalizing Quadratic Forms Conic Sections. 9.7 Quadric Surfaces. 9.8 Comparison of Procedures for Solving Linear Systems. 9.9 LU-Decompositions. Chapter 10. Complex Vector Spaces. 10.1 Complex Numbers. 10.2 Division of Complex Numbers. 10.3 Polar Form of a Complex Number. 10.4 Complex Vector Spaces. 10.5 Complex Inner Product Spaces. 10.6 Unitary Normal, and Hermitian Matrices. Chapter 11. Applications of Linear Algebra. 11.1 Constructing Curves and Surfaces through Specified Points. 11.2 Electrical Networks. 11.3 Geometric Linear Programming. 11.4 The Earliest Applications of Linear Algebra. 11.5 Cubic Spline Interpolation. 11.6 Markov Chains. 11.7 Graph Theory. 11.8 Games of Strategy. 11.9 Leontief Economic Models. 11.10 Forest Management. 11.11 Computer Graphics. 11.12 Equilibrium Temperature Distributions. 11.13 Computed Tomography. 11.14 Fractals. 11.15 Chaos. 11.16 Cryptography. 11.17 Genetics. 11.18 Age-Specific Population Growth. 11.19 Harvesting of Animal Populations. 11.20 A Least Squares Model for Human Hearing. 11.21 Warps and Morphs. Answers to Exercises. Index.
442 citations
Book•
1 Jan 1986
339 citations
Book•
1 Aug 1998
TL;DR: In this paper, the authors present models of linear systems vectors and vector spaces linear operators on vector spaces eigenvalues and eignevectors functions of vector matrices solutions to state equations system stability controllability and observability system realizations state feedback and observers introduction to optimal control and estimation mathematical tables MATLAB command summaries.
Abstract: Models of linear systems vectors and vector spaces linear operators on vector spaces eigenvalues and eignevectors functions of vector matrices solutions to state equations system stability controllability and observability system realizations state feedback and observers introduction to optimal control and estimation mathematical tables MATLAB command summaries.
290 citations
Book•
1 Dec 1982TL;DR: In this article, the authors present a general algebra of vector spaces, which includes linear transformations and characteristic roots, as well as other special matrices such as patterned matrices, idempotent and tripotent matrices and projections.
Abstract: Prerequisite matrix theory. Prerequisite vector theory. Linear transformations and characteristic roots. Geometric interpretations. Algebra of vector spaces. Generalized inverse. Conditional inverse. Systems of linear equations. Patterned matrices and certain other special matrices. Trace of a matrix, vector of a matrix, commutation matrices. Integration and differentiation. Positive matrices and matrices with non-positive off-diagonal elements. Non-negative matrices, idempotent and tripotent matrices, projections.
240 citations
TL;DR: In this paper, it was shown that the optic flow field arising from motion relative to a visually textured plane may be characterized by eight parameters that depend on the observer's linear and angular velocity and the coordinate vector of the plane.
Abstract: It is shown that the optic flow field arising from motion relative to a visually textured plane may be characterized by eight parameters that depend on the observer's linear and angular velocity and the coordinate vector of the plane. These three vectors are not, however, uniquely determined by the values of the eight parameters. First, the optic flow field does not supply independent values for the observer's speed and distance from the plane; it only gives the ratio of these two quantities. But more unexpectedly, the equations relating the observer's linear velocity and the plane's coordinate vector to the eight parameters are still satisfied if the two vectors are interchanged or reversed in direction, or both. So in addition to the veridical interpretation of the optic flow field there exist three spurious interpretations to be considered and if possible excluded. This purpose is served by the condition that an interpretation can be seriously entertained only if it attributes every image element to a light source in the observer's field of view. This condition immediately eliminates one of the spurious interpretations, and exhibits the other two as mutually inconsistent: one of them is tenable only if all the visible sources lie on the forward half of the plane (relative to the observer's linear velocity); the other only if they all lie on the backward half-plane. If the sources are distributed over both halves of the plane, only the veridical interpretation survives. Its computation involves solving a 3 $\times $ 3 eigenvalue problem derived from the flow field. If the upper two eigenvalues coincide, the observer must be moving directly towards the plane; if the lower two eigenvalues coincide, his motion must be directly away from it; in both cases the spurious interpretation merges with the veridical one. If all three eigenvalues are equal, it may be inferred that either the observer's linear velocity vanishes or the plane is infinitely distant.
224 citations
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| 2022 | 2 |
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| 2020 | 15 |
| 2019 | 11 |
| 2018 | 14 |
| 2017 | 13 |