Useful for UGC NET · ASRB NET · ISS
This is the complete study package for Linear Algebra & Linear Models (STS-102) together with its paired practical STS-106. The course builds the matrix theory that the courses that follow assume, and then turns it on the general linear model — the single formulation behind regression, analysis of variance and analysis of covariance alike.
Inner products and what orthogonality means for random variables; the Gram–Schmidt process; orthogonal projection and its projection matrix; generalized and Moore–Penrose inverses with all four Penrose conditions checked; consistency and the general solution of a linear system.
UNIT 2Building the characteristic equation from traces of successive powers; algebraic against geometric multiplicity and what makes a matrix defective; Cayley–Hamilton used to invert without cofactors; and the spectral decomposition, from which every power, the inverse and the square root follow at once.
UNIT 3Reduction to canonical form, rank, index and signature under Sylvester's law; the leading minor test; the Rayleigh quotient, which is what defines principal components; simultaneous reduction of two forms; and the Cauchy–Schwarz and Hadamard inequalities.
UNIT 4The general linear model; which parametric functions are estimable when the design is rank deficient; the Gauss–Markov theorem proved in full; BLUEs characterised by linear zero functions; Aitken's generalized least squares; and multicollinearity with variance inflation factors.
PRACTICALAll fourteen prescribed practicals. One matrix runs through five of them, so the partition, sweep-out, Doolittle, Cayley–Hamilton and spectral routes all check against one another. The R is written without packages, as the course requires.
REFERENCEThe prescribed unit-wise outline for STS-102 and the practical list for STS-106, as printed, with objectives, outcomes and the reading list.
| What is built here | Where it is used |
|---|---|
| Gram–Schmidt and orthogonal transformations | The Helmert transformation in Distribution Theory, Unit 3 |
| Idempotent matrices, rank and trace | Quadratic forms and Fisher–Cochran in Distribution Theory, Unit 4 |
| Spectral decomposition and the matrix square root | Mahalanobis distance and principal components in Multivariate Analysis (STS-202) |
| The Rayleigh quotient and simultaneous reduction | Principal components, discriminant analysis and canonical correlation |
| Estimability and generalized inverses | Every rank-deficient design in Design and Analysis of Experiments |
| Gauss–Markov and Aitken | Weighted least squares and autocorrelated errors in Econometrics |