Prerequisites:

• Stat 343, 345, 347.

• Linear models: first-order theory
• Random vectors and expectation
• Vectors and vector spaces, vector span, subspace
• Residuals and interaction: coset and quotient space
• The statistical units $U$ the vector space $R^U$
• Linear functionals, measures and the dual space
• Functions on the dual space: moment and cumulant generating function
• Legendre transform
• Dual of a subspace; dual of a quotient
• Direct sum and vector span
• Basis vectors; components of a vector; index notation; Change of basis
• Linear transformation; Injective maps and inverse maps
• Matrices and generalized inverse matrices
• Adjoint linear transformation
• Tensor products; cumulants;

• Orthogonality
• Projection and least squares estimation;
• Subspace and orthogonal complement;
• Balanced designs and analysis of variance;
• Dual inner product;
• Inner product on a subspace;
• Inner product on a quotient space;
• Lebesgue measure
• Residuals and REML

• Invertible maps on $\Omega$ permutations
• Injective maps $\Omega$  -> $\Omega'$ the category Inj;
• Surjective maps $\Omage$ to $\Omega'$ the category Surj;
• Representations: Factors and the representation $R^\Omega$
• Sub-representations; quotient representations;
• Natural transformation (link function); homomorphism;
• Product category and tensor product representations;
• Sub-representations; Factorial models; Analysis of variance;
• Homologous factors and diallel-cross models;
• Exchangeability and invariant measures