This class constitutes the mathematical core of Master in Data Science program. It gives a rigorous, structured review of Linear Algebra and Analysis with particular emphasis on applications in Data Science, Computation, and Modeling and a view toward working in high-dimensional or infinite-dimensional vector spaces. The following topics will be covered:
Review of linear algebra: vector spaces, subspaces, linear independence (HJE 1.1, 1.2)
Review of linear algebra (ctd.): replacement and basis exchange theorems, dimension (HJE 1.4.1, 1.5 without full proofs)
Linear transformations: examples, kernel, range (HJE 2.1)
Linear transformations (ctd.): invertability, isomorphisms, rank-nullity theorem (HJE 2.2 with some details skipped, 2.3.1 with some details skipped), matrix representations (HJE 2.4)
Change of basis (ctd.), elementary matrices, row echelon form and reduced row echelon form (HJE 2.7.1, 2.7.2)
Basic and free variables, permutations, determinants (HJE 2.7.3, 2.8.1 without full proofs, 2.8.2, 2.9.1 without proof of Theorem 2.9.1); inner products (HJE 3.1.1, 3.1.2 in parts)
Orthonormal sets, orthogonal projections (HJE, 3.2.1, 3.2.2)
Gram-Schmidt and QR (HJE 3.3); normed linear spaces, operator norm (HJE 3.5)
Norm inequalities (HJE 3.6); adjoints (main points of HJE 3.7), fundamental subspaces theorem (HJE, Theorem 3.8.9)
Application of the fundamental subspaces theorem: least squares (HJE 3.9.1); eigenvalues and eigenvectors (HJE 4.1), diagonalization (HJE 4.3.1);
Schur’s lemma and the spectral theorem for Hermitian and normal matrices (HJE 4.4)
Singular value decomposition (HJE 4.5 and main ideas of 4.6)
Metric spaces, open sets, continuous functions (HJE 5.1 and 5.2.1)
Closed sets, sequences, convergence (HJE 5.2 and 5.3 without some of the proofs/details)
Completeness and uniform continuity (HJE 5.4.1 and 5.4.2)
Compactness (HJE 5.5.1, 5.5.2), every sequence on a compact set has a convergent subsequence (one of the several statements of HJE Theorem 5.5.11), uniform vs. pointwise convergence (HJE 5.6.1)
Banach spaces, continuous linear extensions (HJE 5.6.3, 5.7.1 with only sketch of proof, 5.7.3 without full proofs)
Uniform limits of continuous functions are continuous (HJE 5.6.2); Banach-valued integration (HJE 5.10 with some proofs skipped)
Differentiation: directional derivatives, partial derivatives, Fréchet derivative in finite dimensions (HJE 6.1, 6.2, start of 6.3.1, all formulated in a Banach space setting with \(\mathbb{R}^n\) only as a particular example)
Properties of the Fréchet derivative (parts of HJE 6.3, 6.4)
Mean value theorem and fundamental theorem of calculus (HJE 6.5.1, 6.5.2)
Taylor’s theorem (HJE 6.6 with simplified proofs only for special cases)
Contraction mapping principle (HJE 7.1)
Uniform contraction mapping theorem (HJE 7.2.1, only Lemma 7.2.6 is proved, short sketch of idea of full proof); implicit and inverse function theorem (selected parts of HJE 7.4, no proofs so far, use of a second-order inverse function theorem computation to explore Example 6.28)
Newton’s method (HJE 7.3, mainly vector-valued case, without proof of quadratic convergence)
Connection between implicit and inverse function theorems, examples, general review
Mock Exam (tentative, date may change!): Review sheet for mock and final exam
Multivariate integration in the sense of Daniell-Lebesgue, sets of measure zero (HJE 8.1, 8.2, 8.3.1)
Measurability, monotone convergence, Fatou’s lemma, dominated convergence (HJE 8.3.2, 8.4.2 with proof, 8.5 without full proofs)
Integration on unbounded domains, Fubini-Tonelli, change of variable formula (Examples from HJE 8.4.3, HJE 8.6.1 but stated as a Fubini-Tonelli theorem, HJE 8.7.2, 8.7.3)
Final Exam, 8:00-10:00 in HB-111