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

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Classroom diary (February-June 2025):
 

PART 1:  LINEAR ALGEBRA

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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]

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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. 

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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]

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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]

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