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, pp. 159-171); Metric spaces, open sets, continuous functions and limits (HJE, pp. 179-190)
Closed sets, sequences, convergence (HJE, pp. 190-195)
Completeness and uniform continuity (HJE, pp. 195-201)
Compactness (HJE, pp. 201-208)
Banach spaces, uniform convergence, continuous linear extensions (HJE, pp. 210-217)
Topologically equivalent metrics, homeomorphisms, connectedness (HJE, pp. 219-227)
Banach-valued integration (HJE, pp. 227-233)
Differentiation: directional derivatives, partial derivatives, Fréchet derivative in finite dimensions (HJE, pp. 241-251)
General Fréchet derivative, properties of derivatives (HJE, pp. 252-259)
Mean value theorem and fundamental theorem of calculus (HJE, pp. 260-265)
Taylor’s theorem (HJE, pp. 266-272)
Contraction mapping principle, uniform contraction mapping principle, Newton’s method (HJE, pp. 277-284)
Newton’s method (HJE, pp. 286-293)
Implicit and inverse function theorem (HJE, pp. 293-300)
Multivariable integration (HJE, pp. 319-330)
Measurability, monotone convergence (HJE, pp. 331-340)
Fatou’s lemma, dominated convergence, differentiation under the integral (HJE, pp. 340-349)
Change of variables (HJE, pp. 349-356)
TBA