Linear algebra II, TCD 2018/19
Syllabus:
- Kernels and images, rank and nullity, dimension formula.
- Invariant subspaces. An application: two commuting linear operators have a common eigenvector. Direct sums.
- Characteristic polynomials. Cayley--Hamilton theorem. Minimal polynomial of a linear transformation. Examples.
- Normal form of a nilpotent operator. Jordan normal form (Jordan Decomposition Theorem). Applications: closed expressions for Fibonacci numbers and other recursively defined sequences.
- Orthonormal bases; Gram-Schmidt orthogonalisation. Orthogonal complements and orthogonal direct sums. Bessel's inequality.
- Bilinear and quadratic forms. Sylvester's criterion. The law of inertia. Spectral Theorem for symmetric operators.
Course materials:
| Lecture | Topics covered | Lecture notes/slides | Homeworks/Tutorials/Solutions |
|---|---|---|---|
| 1 (21/01) | Introduction. Recollections from module 1111. Kernels and images of linear maps. Rank and nullity of a linear map. | L1 [PDF] | |
| 2 (21/01) | Rank-nullity theorem. | L2 [PDF] | |
| 3 (22/01) | Sums and direct sums. Dimension of the sum of two subspaces. | L3 [PDF] | HW1 [PDF] |
| 4 (28/01) | Computing intersection of subspaces. Example. | L4 [PDF] | |
| 5 (28/01) | Computing relative bases. Example. Invariant subspaces. Characteristic polynomial of a linear transformation. | L5 [PDF] | HW1 solutions [PDF] |
| 6 (29/01) | First tutorial | HW2 [PDF] T1 [PDF] T1 solutions [PDF] | |
| 7 (04/02) | Invariant subspaces. Two commuting operators have a common eigenvector. Canonical forms for linear transformations that satisfy φ2=φ and φ2=0. | L6 [PDF] | |
| 8 (04/02) | Canonical form for a linear transformation that satisfies φk=0. | L7 [PDF] | HW2 solutions [PDF] |
| 9 (05/02) | Examples of canonical forms for linear transformations that satisfy φk=0. | L8 [PDF] | HW3 [PDF] |
| 10 (11/02) | Handling arbitrary linear transformations: kernels of powers and a direct sum decomposition. Separating eigenvalues. | L9 [PDF] | HW4 [PDF] |
| 11 (11/02) | Jordan decomposition theorem. Examples. | L10 [PDF] | HW3 solutions [PDF] Past midterm questions [PDF] |
| 12 (12/02) | Tutorial on Jordan decomposition. | T2 [PDF] T2 solutions [PDF] | |
| 13 (18/02) | Uniqueness of Jordan forms. Invariants of similar matrices. Two proofs of Cayley-Hamilton theorem. | L11 [PDF] | HW5 [PDF] |
| 14 (18/02) | Powers of a Jordan block and applications of Jordan forms to computing powers of matrices. Examples: linear recurrences, Markov chains, matrix differential equations. | L12 [PDF] | HW4 solutions [PDF] |
| 15 (19/02) | Tutorial on applications of Jordan decomposition. | T3 [PDF] T3 solutions [PDF] | |
| 16 (25/02) | Midterm test. | Midterm questions [PDF] | |
| 17 (25/02) | Midterm test. | ||
| 18 (26/02) | Discussion of the midterm test. | Midterm solutions [PDF] | |
| Study week. | |||
| 19 (11/03) | Euclidean vector spaces. Examples. Orthonormal bases, Gram-Schmidt orthogonalization process. | L13 [PDF] | HW6 [PDF] | 20 (11/03) | Lengths and angles in Euclidean spaces. Cauchy-Schwartz inequality. Orthogonal complements. | L14 [PDF] | HW5 solutions [PDF] |
| 21 (11/03) | Orthogonal direct sums. Bessel's inequality. An application of Bessel's inequality to estimating the sum of inverse squares. | L15 [PDF] | |
| 22 (12/03) | Tutorial class on Gram-Schmidt orthogonalization. | T4 [PDF] T4 solutions [PDF] | |
| March 18 is a holiday in lieu of St Patrick's day. | |||
| 23 (19/03) | Quadratic forms (and a motivation from analysis). Symmetric bilinear forms. Canonical form of a symmetric bilinear form. Law of inertia. Theorems of Jacobi and Sylvester. | L16 [PDF] | HW7 [PDF] HW6 solutions [PDF] |
| 24 (25/03) | Proof of the canonical form theorem and the law of inertia. | L17 [PDF] | HW8 [PDF] |
| 25 (25/03) | Proof of the Jacobi theorem and the Sylvester theorem. Statement of the eigenvalue theorem. | L18 [PDF] | HW7 solutions [PDF] |
| 26 (25/03) | Extremal properties of eigenvalues of symmetric matrices. A real symmetric matrix admits an orthonormal basis of eigenvectors. Signature and eigenvalues. | L19 [PDF] | |
| 27 (26/03) | Tutorial on bilinear and quadratic forms. | T5 [PDF] T5 solutions [PDF] | |
| 28 (01/04) | Orthogonal matrices in 2D and 3D. Hermitian vector spaces. | L20 [PDF] | HW8 solutions [PDF] |
| 29 (01/04) | Adjoint of a linear transformation in a Hermitian vector space. Symmetric (self-adjoint) and unitary transformations. Normal transformations and their diagonalisation. An outline of an application of Euclidean vector spaces to face recognition. | L21 [PDF] | Past exam questions [PDF] |
| 30 (02/04) | Tutorial to practice with past exam questions. | ||
| 31 (08/04) | Discussion of selected practice exam questions. | ||
| 32 (08/04) | Discussion of selected practice exam questions. Revision of the module. | ||
| 33 (09/04) | Revision of the module |
My other materials
You may consult the webpage of the year long linear algebra module I was taught in 2014/15. I kept all lecture notes there. Material might change a little bit in this version of the module.Exam materials: sample papers, actual papers
See course materials, mainly. Work through solutions to home assignments and tutorials, since exam papers will rely on fluency in the respective methods. Generally, check out past papers from the previous 4-5 years here.Textbooks
This module does not follow any particular textbook. When needed, consult "Algebra" by Michael Artin. There will also be online lecture notes for many of the lectures. For more examples and exercises (very useful to get a grip of the material), check out the free online linear algebra book by Jim Hefferon available at this http URL. Finally, more advanced reading for the second half of this module is "Linear algebra problem book" by Halmos.Assessment
The final mark is 60%*final exam mark + 20%*midterm test mark + 20%*home assignments result.