Study guide of topics covered 
ST 705
Monahan textbook chapters covered: 0 - 6

definition of a vector space;
linearly dependent vectors and linearly independent vectors;
spanning set of vectors;
basis of vectors;
subspaces;
linear transformations;
null space and range space of a linear transformation;
dimension of a vector space;
rank of a linear transformation;
relationship between linear transformations and matrix multiplication;
dimension theorem;
matrix multiplication;
properties of matrices;
trace of a matrix;
determinant of a matrix;
eigenvalues of a matrix;
eigenvectors of a matrix;
characteristic polynomial of a matrix;
spectral theorem (for finite dimensional vector spaces);
diagonalizability;
simultaneous diagonalizability;
inner product and inner product spaces;
vector and matrix norms and induced norms;
triangle inequality;
Cauchy-Schwarz inequality;
definition of orthogonality;
general linear model;
vector and matrix derivatives;
sum-of-squared error in the general linear model;
least squares solution;
the normal equations;
results about null spaces and column spaces of design matrices;
least squares predictions and properties;
geometry of least squares solutions;
generalized inverses;
Moore-Pensrose conditions;
Moore-Penrose pseudo-inverse;
singular value decomposition;
projection matrices and properties;
orthogonal projection matrices and properties;
orthogonal decompositions;
orthogonal subspaces;
projection onto the column space of a design matrix;
various expressions for the set of least squares solutions;
reparameterizations;
confounding variables;
orthonormal basis;
Gram-Schmidt orthonormalization process;
definition of an unbiased estimator;
definition of a linear estimator;
definition of a linearly estimable function;
properties of linearly estimable functions;
subspace of linearly estimable functions;
methods to determine if a function is estimable;
linear estimates of reparameterized general linear models;
identifiability of parameters;
further matrix decompositions;
imposing conditions for a unique solution to the normal equations;
linear constraints, estimability of constraints, and properties;
constrained parameter spaces;
properties of constrained estimation;
estimability in a restricted model;
restricted normal equations;
Gauss-Markov model/assumptions;
properties of expectation and covariance;
Gauss-Markov theorem and the best linear unbiased estimator (BLUE);
variance estimation;
misspecification: under-fitting and biases;
misspecification: over-fitting and biases;
mean squared error;
Aitken model and theorem;
generalized least squares;
multivariate generalization of a univariate distribution;
multivariate normal distribution and equivalent definitions;
multivariate moment generating function;
joint normality and independence;
definition of a chi-squared random variable;
properties of the chi-squared distribution;
noncentral chi-squared distribution and properties;
construction of the F distribution from a ratio of independent chi-squares;
construction of the student T distribution;
distributions of quadratic forms of multvariate Gaussian random vectors;
Cochran's theorem;
sufficient and minimal sufficient statistics;
complete sufficient statistics;
minimum variance unbiased estimator (MVUE) in the Gaussian linear model;
maximum likelihood estimator (MLE) in the Gaussian linear model;
definition of a general linear hypothesis;
construction of F-tests for general linear hypotheses;
likelihood ratio test (LRT);
equivalence between the LRT and the F-test for general linear hypotheses;
confidence intervals and multiple comparisons (Bonferroni, Sheffe, Tukey);