# TMA4145 Linear Methods, Fall 2014

### Message board

20.12 | Problems and solutions of the final exam are here English,Solution,Bokmål, Nynorsk. There will be no grades before week 3 in 2015. Thank you very much for a nice semester, it was a pleasure to lecture for you. Have nice well-deserved holidays and celebrations. |
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Old messages

Old messages

18.12 | All solutions to the trial exam that were handed in are graded, many of you have done a really great job! Hopefully I have all graded papers now in my office. If you want to get the feedback come by my office today or tomorrow. If I am not there please send me a message. I am also available tomorrow for the last minute questions. |
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10.12 | Additional hours (Spørretimer) next week are Monday 15-17 (December 15) and Wednesday 10-12 (December 17) in S4. If you want me to do some problems on the blackboard, please send your requests by email (preferably in advance). Everyone who handed in the trial exam by Monday got it approved, you can collect your graded papers from the shelf on the 3rd floor (gruppe 5), I am also happy to answer any questions you have, just come by my office. |
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22.11 | Last week we finished SVD and pseudo inverse (if you find the explanation in the lecture notes too short try to look at you Linear algebra book and do the problems from problem set 13). We also did some old exam problems on Thursday. There will be an extra lecture next Thursday (November 27th), 10.15-12.00 in G1. The plan is to go through August exams from 2013 and 2014, problems 3-5 in both. Please, let me know if you have any suggestions/requests. |
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18.11 | We are almost done with the syllabus. I will review the singular value decomposition and pseudoinverse on Wednesday. Then we will repeat most important concepts and do some problems. If you have any suggestions, please, send me an email. The last questionnaire is on the left, please answer some questions about extra meetings before the exam and course evaluation. The last meeting of the reference group is tomorrow. |
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11.11 | Last week we have proved the Caley-Hamilton theorem and discussed the Jordan normal form. On Wednesday we will continue to study the Jordan normal form and the exponential of a matrix |
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03.11 | Last week Harald went through the section 4.7, we start 4.8 on Wednesday. I will be late for the office hours tomorrow, should be in the office after 10.00. |
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24.10 | We proved the Riesz representation theorem and the existance of the adjoint operator. Some examples of self-adjoint operators were discussed. Next week lectures will start from 4.7. Harald Hanche-Olsen will give the lectures, there will be no office hours next week. |
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22.10 | NB: The Problem set 10 was updated today, the changes are in the last problem. Todays lecture was about bounded functionals and the dual space. We had some examples of bounded functionals and formulated the Riesz representation theorem, the proof is postpone til tomorrow. |
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16.10 | Yestarday we discussed the kernel and the range of a linear operator and the rank-nullity theorem for linear operators in finite dimensional spaces. Today we finished section 4.4 and introduced bounded linear transformations between normed vector spaces, examples with computations of norms were given. |
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15.10 | Updated notes have been uploaded, see the link on the Course material page. |
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10.10 | Yesterday we discussed matrices of linear transformations, LU-factorization, change of basis and similar matrices. We are done with sections 4.1-4.3 of chapter 4, revised version of the notes will be upload before the next lecture. |
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08.10 | We started chapter 4 today, defined linear transformations between vector spaces and operations on linear transformations: addition, multiplication by scalasrs and composition. Linear transformations between inite dimensional spaces with fixed bases a given by matrices. |
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08.10 | There will be a meeting of the referense group today. Please send them your comments and/or answer this week questionaire (link on the left). |
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04.10 | This week we finished chapter three, the main topics were projections, orthogonal systems and bases, Gram-Schmidt orthogonalization algorithm and QR-factorization of matrices. |
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25.09 | On Thursday lecture we first proved the closest point theorem and a corollary and discussed examples when the convex set is a subspace. Then we said when a space is the direct sum of its subspace and proved the projection theorem. |
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24.09 | The topic today was inner product spaces: Examples, Cauchy-Schwarz inequality, norm, Hilbert spaces. We formulated the closest point theorem and will start the lecture tomorrow by proving it. See sections 3.5-3.6 in the Lecture notes. |
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18.09 | We started normed spaces, defined Banach spaces and Schauder bases. Slides from this week lectures are here. Please answer this week quiestionaire (the link is on the left). Office hours next week are on Tuesday 9.00-10.30 |
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17.09 | The lecture was mainly devoted to the repetition of what you know from linear algebra: vector spaces, subspaces, linear combinations, bases. We showed that dimension of a finitely dimensional vector space is well defined and that each finite dimensional vector space is isomorphic to R^{n}. Thank you for answering my questions. |
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11.09 | We worked through Picard's iterations for ODE. It was the last lecture on metric spaces and the Banach fixed point theorem. |
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10.09 | Today we have proved the Banach fixed point theorem and discussed some simple applications (section 2.6 in the lecture notes), we started to look at the application to existence of initial value problem for first order ODE. I tried to go through the proof of completeness of the function space C(I,U) (see section 2.4). |
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04.09 | We have finished section 2.3 of the notes (skipped 2.4, read it by yourself!) and then discussed functions between the metric spaces and contraction. The Banach fixed point theorem was formulated and a number of examples given. The proof is postponed til the next week |
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04.09 | On Wednesday's lecture we looked at the list of standard metric spaces including the sequence spaces and different metrics on the space of continuous bounded functions on a bounded interval. Then we defined Cauchy sequences and complete metric spaces. We concluded the lecture by proving that R^{n} is complete. Thank you very much for your feedback! |
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02.09 | The first meeting of the reference group is on Wednesday, September 3rd at 16.15. Please, give us some feedback, talk to the members of the reference group or write an email to one of them. We also will start with weekly short feedback questionnaires. Hope you take 2 minutes to answer the question, the answers are anonymous and should help me to plan the lectures and exercises sets further. Thank you! |
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28.08 | We finished section 2.2 of the notes which includes a characterization of the closure through sequential limits and the Bolzano-Weierstrass theorem. We will start the Cauchy sequences and completeness next week, please begin reading 2.3 in the lecture notes. |
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27.08 | Many new terms today: balls in metric spaces, inetrior and boundary points, the interior and closureof a set, open and closed subsets of a metric space. We also introduced the space l_{∞}. Then we started with limits and proved that the limit is unique. |
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21.08 | We did some repetition today, discussed the completeness axiom for the real numbers and introduced the metric spaces. Please, read the first chapter and the first page of the second chapter in the lecture notes. We have four volunteers for the reference group, they are from BMAT, MTTK, MTFYMA (two). If you are from another study program or you don't think those can represent you properly by other reasons, please sign up for the reference group by sending an e-mail to me. |
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20.08 | It was great to see many of you on the lecture today! We discussed the cource and started working through the lecture notes, sections 1.1-1.5. Main terms from today are: set, element, subset, intersection, union, complement, function. We continue with functions tomorrow, discuss real numbers and start metric spaces. If you have any questions, step by my office on Thursday! |
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03.08 | First lecture is on Wednesday, August 20th. There is no need to buy a book for this course, lecture notes will be available. If you still have Linear Algebra book you bought for Calculus 3/Linear algebra courses, don't get rid of it yet. |
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### Time and place

Lectures are held **Wednesdays 12.15, G1, and Thursdays 14.15, S4**. There will be one problem session each week, starting the week August 25-29. For details, see the schedule for TMA4145.

**Course contents and preliminary lecture plan**

Preliminary lecture plan is here (klick).

Preliminary lecture plan is here (klick).

We. 20.08 | 1. Introduction. Lecture notes sections 1.1–1.5 (Sets; Membership and inclusions; Set operations; Relations). | Friedberg et al. 549–552; Kreyszig 609–617. |
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Th. 21.08 | 2. Lecture notes sections 1.6, 2.1 (Real numbers; Metric spaces- first examples) | Friedberg 553–555, Kreyszig 1.1 |

We. 27.08 | 3. Lecture notes section 2.2 ( Metric spaces, Neighbourhoods, Sequences, Limits, Continuity)) | Kreyszig 1.2-1.3 |

Th. 28.08 | 4. Lecture notes section 2.3 (Cauchy sequences, Limits, Completeness, Bolzano-Weierstrass theorem) | Kreyszig 1.3-1.4 |

We. 03.09 | 5. Lecture notes section 2.4-5 (Completeness, Examples; Functions between metric spaces) | Kreyszig 1.5. |

Th. 04.09 | 6. Lecture notes section 2.6 (Contractions, Banach fixed point theorem) | Kreyszig 5.1-2 |

We. 10.09 | 7. Lecture notes section 2.7 (Banach fixed point theorem, Applications) | Kreyszig 5.3-4 |

Th. 11.09 | 8. Lecture notes sections 3.1-2 (Linear subspaces; linear dependence; Bases and dimension) | Friedberg et al: 1.3–1.6; Kreyszig 2.1 (2.4); Strang 2.1, 2.3. |

We. 17.09 | 9. Lecture notes sections 3.3-4 (Normed spaces, Schauder bases) | Kreyszig 2.2-3; Strang 2.1, 2.3. |

Th. 18.09 | 10. Lecture notes 3.5-3.6 ( Inner-product spaces and Hilbert spaces) | Kreyszig 3.1, Young 1.1, 3.1, Friedberg et al. 6.1 |

We. 24.09 | 11. Lecture notes 3.6 ( Convex sets and the closest point property) | Kreysig 3.3, Young 3.2 |

Th. 25.09 | 12. Lecture notes 3.7 ( Orthogonality, the projection theorem) | Kreyszig 3.3, Strang 3.1 |

We 01.10 | 13. Lecture notes section 3.8 (Orthonormal sequences, Bessel's inequality. Fourier coefficients) | Kreyszig 3.4-6, Young 4.1, 4.3 |

Th 02.10 | 14. Lecture notes 3.8 (Gram-Schmidt orthogonalization and QR-decomposition) | Kreyszig 3.4, Strang 3.4, Friedberg et al. 6.2 |

We. 08.10 | 15. Lecture notes section 4.1 (Linear transformations and matrices) | Kreyszig 2.6, Friedberg et al. 2.1, 2.3 |

Th. 09.10 | 16. Lecture notes sections 4.2, 4.3 (Gaussian elimination and LU decomposition, change of basis) | Friedberg et al: 3.4 (especially 183–188); Strang 1.5, 2.2. |

We. 15.10 | 17. Lecture notes 4.4 (Kernels and ranges of linear transformations) | Friedberg et al. 2.1; Kreyszig 2.6 and 2.9; Strang 2.2, 2.4 and 2.6. |

Th. 16.10 | 18. Lecture notes section 4.5 (Bounded linear transformations) | Kreyszig 2.7 |

We. 22.10 | 19. Lecture notes section 4.5 (Bounded functionals, the dual space) | Kreyszig 2.7 |

Th. 23.10 | 20. Lecture notes section 4.6 (Riesz representation theorem, adjoints) | Friedberg et al: 6.3; Young 6.1; Kreyszig 3.8, 3.9 |

We. 29.10 | 21. Lecture notes section 4.7 (Functional calculus: exponential of an operator) | Kreyszig 7.1 |

Th. 30.10 | 22. Lecture notes section 4.8 (Spectum, invariant subspaces) | Kreyszig 7.1, Friedberg et al. 5.1, 5.4 |

We. 05.11 | 23. Lecture notes section 4.8 (Caley-Hamilton theorem) | Friedberg et al. 5.4 |

Th. 06.11 | 24. Lecture notes 4.8 (The Jordan form, applications) | Friedberg et al. 7.1-7.2 |

We. 12.11 | 25. Lecture notes 4.8 (Spectral theorem for hermitian matrices) | Young 7.3, Friedberg et al. 6.2-6.4, Strang 3.4, 5.2. |

Th. 13.11 | 26. Lecture notes 4.8 (Positive definite matrices and SVD-decomposition.) | Young et al. 8.2. Friedberg et al. 6.5-6.8, Strang 6.1-6.3. |

We. 19.11, Th. 20.11 | Repetition. |

### Lecturer

Eugenia Malinnikova

Homepage. My office is 948, 9th floor, SBII.

Office hours: Tuesday 9-11 or by appointment

Please contact me at eugenia [at] math [dot] ntnu [dot] no
if you have any questions concerning the course.

### Reference group

Paul Trygsland (MTFYMA) Patrygsl [at] stud [dot] ntnu [dot] no

Harald Sperre (MTFYMA) haralesp [at] stud [dot] ntnu [dot] no

August Peter Bridelen Sonne (BMAT) Apsonne [at] stud [dot] ntnu [dot] no

Øyvind Harding Guldbrandsen (MTTK) oyvindhg [at] gmail [dot] com

Minutes from the first meeting, September 3rd (klick).

Minutes from the first meeting, September 3rd (klick).

Regarding the lectures: - Most students seem to find the lectures engaging and are mostly happy with them. - Some are unfamiliar with the notation, definitions and proofs, and need repetition. However, some find repetition tedious. - It is recommended not to note everything that is written on the blackboard. Most of it is already written in the lecture notes.

Suggestions for upcoming lectures: - Make it clearer when proofs are finished. Can for instance write "qed" on the blackboard. - Writing the number of the current chapter makes it easy to compare your own notes to the lecture notes. - More headlines makes it more organized, e.g. "Definition:", "Proof:" - Let new definitions "sink in". Allow for small pauses. - Draw more pictures for explaining, it makes abstract concepts easier to grasp.

Regarding the problem sets: - Problems in the problem sets are about equally as difficult as the problems on the exam - Many would like the first problem in the sets to be a little more straight-forward, so that one does not get discouraged by the first task.

Other: - Feedback questioning will continue. - Could be a good idea to make list that translates math terms between Norwegian and English.

Minutes from the second meeting, October 8th (klick).

Minutes from the second meeting, October 8th (klick).

Feedback:

- Some students feel like the lectures are far from the problem sets. - Students find the lectures good, and find it easier to follow now. More headlines, students are more adjusted etc…

Lecture Notes:

- Lecturer and students are not satisfied with last part of chapter 3. A new version will (hopefully) be uploaded. -Lecturer want to take a different approach to chapter 4 aswell, and will do her best to upload a new version within the next week. - The Lecture Notes is a work in progress, unfortunately there will be no final version before the exam. (Note that this process can take several years.)

Problem sets:

- Students are more comfortable with the problem sets. Proofs are hard, more hints would be good, as it isn't always easy to be creative. - Students feel like the exam problems are alot easier than the other exercises. -There are some variation in the degree of difficulty on the problem sets. For instance problem set 7 is regarded as alot harder than problem set 5 and 6.

Exam:

- What to do between final lecure and exam? Some ideas: Guidelines on how to study, list of important topics to repeat with relevant exam tasks. - Students should note that the relevance of previous exams varies. - Finally there will be a "spørretime" about a week before the exam; Anonymous questions, questions, help from lecturer etc…

Other:

- Still not many people attending office hours. - A Norwegian-English dictionary would still be a good idea.

### Exam

The exam is to take place on the 20th of December.