6 hours per week, 3 days a week. There are three sessions per week:
This course offers a deeper study of linear algebra, combining computational techniques with abstract theory. Students begin with orthogonality, projections, least‑squares problems, block matrix methods, and the structure of symmetric and positive definite matrices.
The course develops the spectral theorem, the singular value decomposition (SVD), and introduces complex vector spaces, Hermitian and unitary matrices, the discrete Fourier transform (DFT), and the fast Fourier transform (FFT).
The second part explores linear operators on general inner product spaces, including adjoints, self‑adjoint and normal operators, and abstract diagonalization. It concludes with generalized eigenvectors, nilpotent operators, and the construction of Jordan normal form.
Apply advanced orthogonality and least‑squares techniques: Use orthogonal projections, QR factorization, and least‑squares approximations. Interpret geometric meaning of orthogonality and approximation.
Analyze key structural classes of matrices: Recognize and work with symmetric, orthogonal, normal, and positive definite matrices. Use Schur complements, block matrix operations, and spectral characterizations.
Handle central matrix decompositions: Compute and interpret QR factorization, spectral decomposition, and SVD. Apply low‑rank approximations and understand geometric transformations of linear maps.
Work fluently with complex vector spaces: Manipulate complex vectors, Hermitian and unitary matrices. Understand complex inner products and properties of complex linear transformations. Apply the discrete Fourier transform (DFT) and understand the structure of the FFT algorithm.
Transition from matrix computations to abstract operator theory: Understand adjoint, normal, unitary, and self‑adjoint operators.
Understand generalized eigenstructure and canonical forms: Construct generalized eigenvectors, nilpotent blocks, and Jordan chains. Determine the Jordan Normal Form and interpret its relation to the minimal polynomial.
There will be one final exam on the dedicated exam week at the end of the semester. During the semester, there will be two more partial exams on dates to be defined (the dates in the table are tentative). There will also be quizzes, homework or presentations that will provide points, out of which a final grade will be computed for the quizzes component of the grade.
The final grade will be computed on the following way:
| Evaluation | Date | Weight |
|---|---|---|
| Partial exam 1 | 2026-03-16 (Monday) | 25% |
| Partial exam 2 | 2026-05-12 (Tuesday) | 25% |
| Quizzes | All semester | 10% |
| Final exam | Final exam week | 40% |
After the second partial exam, here is the plan for the upcoming weeks listed by session:
Axler, Chapter 4, Polynomials
Tuesday, May 19, 14h: - Proper proofs regarding polynomials over the fields $\mathbb{C}$ and $\mathbb{R}$ - 4.6: each zero of a polynomial corresponds to a degree-one factor - 4.8: degree $m$ implies at most $m$ roots - 4.9: division algorithm for polynomials - Factorization over $\mathbb{C}$ - 4.13: fundamental theorem of algebra, second version. First version will not be proved in class. - Factorization over $\mathbb{R}$ - 4.14: complex roots of real polynomials come in pairs - 4.15: factorization of quadratic polynomials - 4.16: factorization of polynomials over $\mathbb{R}$
Wednesday, May 20, 09h:
Exercises from Chapter 4 of Axler. Some of them include definitions that we have not seen in class: those can be ignored.
Suggested exercises include: 4-11, 14
If there is extra time, it will be used to review Chapter 4 with students depending on their questions. If there are no questions, then we will start the courses of next week. In particular, the topics of next session (Section 5A) are very abordable by now.
Axler, Chapter 5, Eigenvalues and Eigenvectors.
This might sound repetitive, but as you will see, the perspective in this book is very different from the one we have studied so far: here, the main idea is understanding an operator using invariant subspaces.
Tuesday, May 26, 14h: Section 5A: Invariant subspaces. This includes the definition of invariant subspace, eigenvalue and eigenvector (review). Second part includes polynomials of operators. If time allows it, we can cover the first subsection of 5B about the existence of eigenvalues, before introducing the minimal polynomial.
Wednesday, May 27, 09h: Section 5B: The minimal polynomial. - 5.19: existence of eigenvalues (in a complex vector space) - 5.22: existence, uniqueness and degree of minimal polynomial - 5.27 eigenvalues are the roots of the minimal polynomial - 5.29: polynomials on $T$ and the minimal polynomial - 5.31: minimal polynomial of a restriction over an invariant subspace - 5.32: invertibility test in terms of the minimal polynomial - 5.33: on real, even-dimensional vector spaces - 5.34: eigenvalues exist on odd-dimensional vector spaces
If time allows it, we will do exercises from Sections 5A and 5B
Axler, Chapter 5, Eigenvalues and Eigenvectors, continued. This week, we will finish this chapter.
Monday, June 01, 15h45: Section 5C: Upper-triangular matrices - The section starts with a review of the matrix of a linear transformation, and types of matrices: diagonal, triangular. - Interesting part begins: - 5.39: conditions for upper-triangular matrix, link with 6.37 that adds that the basis can be made orthonormal - 5.40: upper-triangular form and polynomials of $T$ - 5.41: eigenvalues of a triangular matrix are in the diagonal - 5.44: characterization of transformations with upper-triangular matrix in terms of the minimal polynomial - 5.47: Over $\mathbb{C}$, every operator has an upper-triangular matrix (Schur decomposition in 6.38 when adding the orthonormal condition to the basis)
Tuesday, June 02, 14h:
Section 5D: Diagonalizable Operators. Many things in this chapter have already been covered: we go over them as review, and then prove again some results using the new approach. - 5.54: sum of eigenspaces is a direct sum - 5.55: equivalent conditions to diagonalizability - 5.62: necessary and sufficient conditions for diagonalizability using the minimal polynomial - 5.65: restriction of a diagonalizable operator to invariant subspace is diagonalizable
Wednesday, June 03, 09h:
Section 7A: Self-adjoint and normal operators - 7.1: definition of adjoint as a function - 7.4: adjoint of a linear map is a linear map - 7.5: properties of the adjoint - 7.6: Null space and range of the adjoint - 7.9: the matrix of the adjoint is the conjugate transpose - 7.10: definition of self-adjoint (Hermitian) - 7.12: eigenvalues of self-adjoint operators - 7.13-7.16: misc results - 7.20: characterization of normal operators in terms of norm - 7.21: range, nullspace and eigenvectors of a normal operator - 7.22: orthogonality of eigenvectors of a normal operator - 7.23: normality and commutativity of real and imaginary parts
Section 7B: Spectral theorems - Real spectral theorem - 7.26: invertible quadratic expressions - 7.27: minimal polynomial of self-adjoint operators - 7.29: real spectral theorem - Complex spectral theorem - 7.31: complex spectral theorem
Axler, Chapter 8, Operators on Complex Vector Spaces. The idea here is to discuss Nilpotent operators, generalized eigenvectors and Jordan form
Monday, June 08, 15h45: Section 8A: Generalized Eigenvectors and Nilpotent Operators - 8.1: sequence of increasing null spaces - 8.2: equality in the sequence of null spaces - 8.3: null spaces stop growing - 8.4: V as a direct sum: similar to rank-nullity theorem - 8.8: definition of generalized eigenvectors - 8.9: there always exist a basis of generalized eigenvectors - 8.11: generalized eigenvector corresponds to a unique eigenvalue - 8.12: linearly independent generalized eigenvectors - 8.16 and 8:17: nilpotent operators - 8.18: minimal polynomial and upper-triangular matrix of nilpotent operator
Tuesday, June 09, 14h: Section 8B: Generalized Eigenspace Decomposition - 8.22: generalized eigenspace decomposition - 8.25: multiplicities of eigenvectors - 8.26,8.28: characteristic polynomial - 8.29: Cayley-Hamilton theorem - 8.30: characteristic polynomial and minimal polynomial - 8.31: more on the multiplicities of eigenvalues - 8.37: block diagonal matrix with upper-triangular blocks
Monday, June 15, 15h45: 8C Consequences of Generalized Eigenspace Decomposition, focus on Jordan form - 8.44: Jordan basis - 8.45: Jordan basis for nilpotent operators - 8.46: Jordan form
Tuesday, June 16, 15h45: Section 8D: Trace: A Connection Between Matrices and Operators
Wednesday, June 17, 09h: Most likely, we will need this time to cover some of the topics that we were not able to cover on the designated time. If not, then review and exercises.