Fund_alg_an_II_24/25 | Paolo Dolce

References:
[1] Kostrikin, Manin - Linear Algebra and Geometry

[2] Zorich - Mathematical Analysis I

[3] Zorich - Mathematical Analysis II

[4] Do Carmo - Differential Forms and Applications

Classroom diary (February-June 2025):

PART 1: LINEAR ALGEBRA

1 - 17/02 - Tensor product of R-modules. Associativity and commutativity of the tensor product (as exercises). Tensor product of vector spaces. Ref. [1, Chap. 4.1, 4.2]
2 - 19/02 - Tensor products and dual vector spaces. Tensor product of linear maps. Extension of scalars: R-modules and vector spaces. Complexes of R-modules. Short exact sequences. Ref. [1, Chap. 4.2, 4.3]

3 - 21/02 - On the (non)-exactness of the tensor product. The tensor algebra. The exterior algebra. Exterior powers and wedge product of vectors. Ref. [1, Chap. 4.3, 4.6]

4 - 24/02 - Exterior powers and alternating maps. Basis of the exterior power of a vector space. Determinant of a vector space and determinant of a linear map. The determinant is non-zero if and only if the linear map is invertible. The determinant of the composition is the product of determinants. Ref. [1, Chap. 4.6]

5 - 26/02 - The determinant of a linear map is equal to the determinant of any matrix representation. Basic properties of the determinant of a matrix. Leibniz expansion of the determinant (i.e. the formula involving the permutation group). The determinant of the transpose. The determinant is invariant up to sign during the Gauss algorithm. Laplace expansion of the determinant.

6 - 28/02 -  Matrix representation of linear maps with respect to special bases in the domain and codomain. Invariant subspaces. Diagonalizable matrices. Eigenvalues and eigenvectors. Ref. [1, Chap. 1.8]
7 - 03/03 -  Characteristic polynomial, eigenvalues and roots of the characteristic polynomial. Jordan blocks and Jordan matrices. Ref. [1, Chap. 1.8]
8 - 05/03 - The minimal polynomial and its properties. The adjugate matrix. Cayley-Hamilton theorem. Root vectors.  Ref. [1, Chap. 1.8]

9 - 07/03 - Root subspaces (i.e.  subspaces associated to a root vector). Decomposition of a vector space into direct sum of root subspaces.  Ref. [1, Chap. 1.8]
10 - 10/03 - Nilpotent linear maps. Jordan basis form for nilpotent maps.  Ref. [1, Chap. 1.8]
11 - 12/03 -  Jordan canonical form. Characterization of diagonalizable matrices. Fundamental theorem of algebra  Ref. [1, Chap. 1.8]

PART 2: GEOMETRY
12- 14/03 -  Bilinear forms and Gram matrices. Inner products (symmetric, skew-symmetric, hermitian). Orthogonality. The kernel of a bilinear form  Ref. [1, Chap. 2]
28 - 23/04 (part 2) - Bilinear forms and associated linear maps. Isometries. Ref. [1, Chap. 2]

29 - 25/04 - Classification of low dimensional inner product spaces. Ref. [1, Chap. 2]

30 - 27/04 - Orthogonal complement of a subspace. Orthogonal completion of a basis. Double orthogonal complement. Ref. [1, Chap. 2]

31 - 30/04 - Orthogonal decomposition of inner product space. Signature of inner product spaces. The signature determines the space up to isometry (case: symplectic and complex symmetric). Ref. [1, Chap. 2]

32 - 07/05 - The signature determines the space up to isometry (case: hermitian and real symmetric). Orthogonal and orthonormal bases. Symplectic bases. Canonical Gram matrices with respect to orthonormal bases. Quadratic forms. Gram-Schmidt algorithm. Ref. [1, Chap. 2]

33 - 09/05 - Euclidean spaces and angles. Orthogonal projections and distances. Volumes of parallelopipeds and determinants. Ref. [1, Chap. 2]
34 - 10/05 - Unitary spaces. Complexification and decomplexification. Euclidean space associated to a unitary space. Ref. [1, Chap. 2 & 1.12]

35 - 12/05 - More on unitary spaces. Characterization theorem for isometries. Orthogonal and unitary operators. Ref. [1, Chap. 2]

36 - 14/05 - Orthogonal and unitary groups. Classification of the elements O(2). Characterization of orthogonal and unitary operators in terms of their matrix representations. Ref. [1, Chap. 2]

PART 3: ANALYSIS
13- 17/03 - Tangent bundle of R^n. Differential k-forms and basic operations between forms Ref. [4, Chap. 1]

14- 19/03 - Differentials are 1-forms. Wedge product of diff. forms. Pullback of diff. forms and coordinate expression. Ref. [4, Chap. 1]

15 - 21/03 - Properties of the pullback of diff. forms. Exterior differentiation and properties. Ref. [4, Chap. 1]

16 - 24/03 - Line integrals of 1-forms. Physical interpretation of line integrals. Ref. [4, Chap. 2]
17 - 26/03 - Complements of measure theory I: \sigma-finite measures. Product of \sigma-algebras. Dynkin systems. Measure of vert./hor. section of measurable sets.

18 - 28/03 - Complements of measure theory II: Existence and uniqueness of the product measure for \sigma-finite spaces. Lebesgue measure on R^n.
19 - 31/03 - Complements of measure theory III: Products of Borel \sigma algebras. Pushforward measures. The Lebesgue measure is invariant under translation. Fubini-Tonelli theorem (start of the proof).

20 - 02/04 - Complements of measure theory IV: Conclusion of the proof of Fubini-Tonelli theorem. Iterated integrals for L^1 functions on products. Double integrals for special domains. Failure of iterated integrals for measurable functions non L^1.
21 - 07/04 - Change of variables for Lebesgue integral (proof postponed) and examples. The integral of 1-forms is invariant (up to sign) under reparametrization of curves. Closed forms and exact forms. A connected open set of R^n is path connected for paths of class C^1 piecewise. Ref. [4, Chap. 2]

22 - 09/04 - Exact forms and invariance of the line integral. Poincare lemma. Example of a diff. form which is locally exact but not exact. Ref. [4, Chap. 2]

23 - 11/04 - Integral of closed forms along paths. Homotopy of paths. Lebesgue numbers of open coverings. The line integral of closed forms is homotopy invariant (start of the proof) Ref. [4, Chap. 2]
24 - 14/04 - Integral of closed forms is homotopy invariant. Freely homotopic loops. Simply connectedness. In a simply connected space 1-forms are exact. Ref. [4, Chap. 2]

25 - 18/04 - Angle functions. Winding numbers and relation with integrals of forms. Free homotopies and winding numbers. Index of a vector field of R^2 Ref. [4, Chap. 2]

26 - 19/04 - Index of a vector field of R^2 and its zeros. Kronecker index formula. Ref. [4, Chap. 2]

27 - 21/04 - Length of a path. Example of a path having infinite length. C^1-piecewise curves have finite length.

28 - 23/04 (part 1) - Curvilinear parameter of a curve. Natural parametrization of a curve. Curvilinear integral of scalar functions and its geometric meaning