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Source document. This page reproduces the syllabus this course was written to, as published — its semesters, credits and paper numbers are that document’s, not this site’s. The course itself is studied on its own, in any order.

This page reproduces the prescribed outline for both papers, so that the teaching pages can be checked against it line by line. It is the syllabus, not a summary of it. Nothing about marks, duration or examination pattern appears on this site.

Course Objectives

  1. Perfection in understanding linear algebraic computations, like determinants, eigen values, eigen vectors, quadratic forms, orthogonalization, matrix transformations, inverses, system of equations solvation, concept of generalized inverses.
  2. Able to understand the real time applications of various concepts of linear algebra and its usage in the statistical theory and must be in a position to interpret.
  3. Understanding the General Linear Models (GLM) and estimation of parameters using GLM.

Course Outcomes

  1. Able to evaluate and interpret the determinants, eigen values, eigen vectors, transformed matrices, orthogonal matrices, solving system of equations, inverse of a matrix.
  2. Able to understand and interpret the results of matrices concepts.
  3. Able to understand the real time applications of the various concepts of matrices.
  4. Able to estimate the parameters of a GLM and its applications in various fields of Statistics.

Unit I

AS PRESCRIBED

Linear Algebra: Vector spaces with an inner product, Gram–Schmidt orthogonalization process. Ortho-normal basis and orthogonal projection of a vector. Real time applications of orthogonalization in various domains. Moore–Penrose and generalized inverses and their properties. Real time applications of solving sets of equations in various domains. Solution of matrix equations. Sufficient conditions for the existence of homogeneous and non-homogeneous linear equations.

→ Unit 1 notes

Unit II

AS PRESCRIBED

Characteristic roots and vectors, Cayley–Hamilton theorem, algebraic and geometric multiplicity of a characteristic root and spectral decomposition of a real symmetric matrix. Real time applications of characteristic roots and vectors in various domains.

→ Unit 2 notes

Unit III

AS PRESCRIBED

Real quadratic forms, reduction and classification of quadratic forms, index and signature. Simultaneous reduction of two quadratic forms, extreme of a quadratic form. Matrix inequalities: Cauchy–Schwartz and Hadamard inequalities.

→ Unit 3 notes

Unit IV

AS PRESCRIBED

Linear Models: General Linear Model (GLM) and its formulation through examples. Estimability of a linear parametric function. Gauss–Markov linear model, BLUE for linear functions of parameters, relationship between BLUEs and linear zero functions. Gauss–Markov theorem, Aitken's generalized least squares, concept of multi-collinearity. Importance and applications of GLMs.

→ Unit 4 notes

Practical Paper STS-106 — Objectives and Outcomes

  1. Knowing the computational procedures and also their implementation using R.
  2. Finding the inverse of a matrix using various methods.
  3. Applying any transformations on matrices.
  4. Applying the matrix operations on the given data sets (determinant, eigen values, eigen vectors, transformations etc).
  5. Summarization of properties of the data sets based on matrix operations.

Practical Paper STS-106 — List of Practicals

  1. Inverse of a matrix by Partition method.
  2. Solutions of linear equations by sweep-out method.
  3. Solutions of linear equations by Doolittle Method.
  4. Computation of Moore–Penrose inverse by Penrose method.
  5. Computation of generalized inverse of a matrix.
  6. Formation of characteristic equation by using traces of successive powers.
  7. Spectral decomposition of a square matrix of third order.
  8. Simultaneous reduction of a pair of quadratic forms to diagonal and canonical forms.
  9. Finding orthonormal basis by Gram–Schmidt process.
  10. Computation of variance-covariance matrix for a data set and study of its characteristics.
  11. Fitting of a simple linear regression model, testing its lack of fit, and computing its R², Adj R², pure error and confidence interval for the regression coefficient.
  12. Fitting of a multiple linear regression model, testing its lack of fit, and computing its R², Adj R², pure error and confidence interval for the regression coefficient.
  13. Computation of simple, partial and multiple correlation coefficients.
  14. Testing multi-collinearity.

The paper's own note requires that the R programs be written without using packages. The semester-end practical examination has two sections: Section A conventional and Section B using R. → Practical notes, both sections

References

  1. Graybill, F. A. (1983): Matrices with Applications in Statistics, 2nd ed., Wadsworth.
  2. Searle, S. R. (1982): Matrix Algebra Useful for Statistics, John Wiley & Sons.
  3. Searle, S. R. (1971): Linear Statistical Models.
  4. Rao, C. R. and Mitra, S. K. (1971): Generalized Inverse of Matrices and its Applications, John Wiley & Sons.
  5. Rao, A. R. and Bhimasankaram, P. (1994): Linear Algebra, 3rd edition, Tata McGraw Hill Publishers.
← Linear Algebra & Linear Models Unit 1 →