Gaussian Elimination" (see Menu) Enter the matrix of coefficients and right-hand side vector You may edit the following statements or use the matrix and vector pallettes to enter new data ( see View‚ Palettes) > A:=<<4 | 2 | 3 | 2> ‚ <8 | 3 | -4 | 7> ‚ <4 | -6 | 2 | -5>>; > b:=<<15‚ 7‚ 6>>; Form the augmented matrix and solve The Maple routine GaussianElimination requires the augmented matrix A|b as input. In this worksheet this matrix is called Ab and is formed using <A|b> >
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[pic] the best (minimum variance) linear (linear functions of the [pic]) unbiased estimator of [pic]is given by least squares estimator; that is‚ [pic]is the best linear unbiased estimator (BLUE) of [pic]. Proof: Let [pic]be any [pic]constant matrix and let [pic]; [pic] is a general linear function of [pic]‚ which we shall take as an estimator of [pic]. We must specify the elements of [pic]so that [pic]will be the best unbiased estimator of [pic]. Let [pic] Since [pic] is known‚ we must find
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13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Matrices—Two–Dimensional Arrays 13 16.1 Size of a matrix . . . . . . . . . . . . 14 16.2 Transpose of a matrix . . . . . . . . 14 16.3 Special Matrices . . . . . . . . . . . 14 16.4 The Identity Matrix . . . . . . . . . 14 16.5 Diagonal Matrices . . . . . . . . . . 15 16.6 Building Matrices . . . . . . . . . . . 15 16.7 Tabulating Functions . . . . . . . . . 15 16.8 Extracting Bits of Matrices
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1.1 SOLUTIONS Notes: The key exercises are 7 (or 11 or 12)‚ 19–22‚ and 25. For brevity‚ the symbols R1‚ R2‚…‚ stand for row 1 (or equation 1)‚ row 2 (or equation 2)‚ and so on. Additional notes are at the end of the section. 1. x1 + 5 x2 = 7 −2 x1 − 7 x2 = −5 1 −2 5 −7 7 −5 x1 + 5 x2 = 7 Replace R2 by R2 + (2)R1 and obtain: 3x2 = 9 x1 + 5 x2 = 7 x2 = 3 x1 1 0 1 0 1 0 5 3 5 1 0 1 7 9 7 3 −8 3 Scale R2 by 1/3: Replace R1 by R1
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Math 111 Homework 1 fall 2007 due 14/9 1. (1.2; 17) Determine the values of h such that the matrix is the augmented matrix of a system which admits a solution. 2 3 4 6 h 7 2. (1.2; 12) Find the general solutions of the system whose augmented matrix is 1 −7 0 6 5 0 0 1 −2 −3 −1 7 −4 2 7 1 −2 4 3. (1.3; 17) Let a1 = 4 ‚ a2 = −3 ‚ b = 1 . For what −2 7 h value(s) of h is b in the plane spanned by a1 and a2? 4. (1.4; 15) Let A = b1 2 −1 and b = . Show that the equation
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.. 15 Matrices .................................................................................................................................................... 27 Matrix Arithmetic & Operations .............................................................................................................. 33 Properties of Matrix Arithmetic and the Transpose ................................................................................. 45 Inverse Matrices and Elementary Matrices ............
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degree of Bachelor of Commerce (Honours) in Economics‚ The University of Auckland‚ 2011. Abstract This dissertation‚ by making use of important geometric and econometric concepts such as a ‘linear manifold’‚ a ‘plane of support’‚ a ‘projection matrix’‚ ‘linearly estimable parametric functions’‚ ‘minimum variance estimators’ and ‘linear transformations’‚ seeks to explore the role of a Concentration Ellipsoid as a geometric tool in the interpretation of certain key econometric results connected
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Linear Least Squares Suppose we are given a set of data points {(xi ‚ fi )}‚ i = 1‚ . . . ‚ n. These could be measurements from an experiment or obtained simply by evaluating a function at some points. You have seen that we can interpolate these points‚ i.e.‚ either find a polynomial of degree ≤ (n − 1) which passes through all n points or we can use a continuous piecewise interpolant of the data which is usually a better approach. How‚ it might be the case that we know that these data points should
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and classes Key OpenCV Classes Point_ Point3_ Size_ Vec Matx Scalar Rect Range Mat SparseMat Ptr Template 2D point class Template 3D point class Template size (width‚ height) class Template short vector class Template small matrix class 4-element vector Rectangle Integer value range 2D or multi-dimensional dense array (can be used to store matrices‚ images‚ histograms‚ feature descriptors‚ voxel volumes etc.) Multi-dimensional sparse array Template smart pointer class
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All the Mathematics You Missed Beginning graduate students in mathematics and other quantitative subjects are expected to have a daunting breadth of mathematical knowledge‚ but few have such a background. This book will help students see the broad outline of mathematics and to fill in the gaps in their knowledge. The author explains the basic points and a few key results of the most important undergraduate topics in mathematics‚ emphasizing the intuitions behind the subject. The topics include linear
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