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TMA4145 Linear Methods

Message board

 Addendum The continuation exam: Bokmål $\:$ Nynorsk $\:$ English $\:$ (No solutions) The results from the exam have been delivered to the department office. Statistics: 130 students met for the exam and 114 delivered solutions; 89 passed and 12 were given the highest grade. Note: Considerations have been given to that Problem 2b has often been misunderstood. Any motivations for the grades ('begrunnelse') will be delivered in person at my office. Exams and solutions: Bokmål $\:$ Nynorsk $\:$English $\:$Solutions I have updated the Themes page. This list is intended as a help for you, but do not forget that the formal curriculum is bigger. Also – do not forget to look at the problem sets. A 'rescue set' 13 has been posted under Problem sets. Karin's office hours are now also posted (here). The time schedule for groups 1–4 is held until the end of November. You can use this opportunity for any questions related to the course (students in group 5 can pick a group of choice). Note, however, that my office hours for Tuesday 20 November (but not for Tuesday 27 November) are cancelled due to a research visit. Lecture notes essentially finished. Last week's lectures will be devoted to repetition (the big picture). There was a typo in an earlier version of problem set 12: The set $X$ in problem 3a should be defined on the interval $[-\frac{1}{2}, \frac{1}{2}]$. You can find some of the earlier exams and problem sets here. Problem set 7 is a mixture of some old exam problems (you may need to look at the latest posted lectures notes, which will be up Friday, Oct. 5). Take this as a real exercise: be thorough and precise, write clearly, and see to that you understand enough of the problems to have passed at the level you would want. Later, when comparing your solutions with the suggested solutions, look for the differences – there will always be similarities, but it is the differences that can help you improve. An earlier version of Problem set 5 contained a typo. In the last problem the coefficient should be $3/4$, without any minus sign. (edited 02.10) The lecture on Thursday, 20.09, will be held by Richard Williamson (please note that my office hours for this week are cancelled). Time for a first checkup. Do you know the concepts on the Themes page? If not, look them up. Go back to the first problem set – do you understand the solutions? If not, ask. Finally, see if you can solve some of the first problems from the old exams. Repetition is the sole way forward. I have updated the course material with chapters, so that those of you who like can read ahead. There is now also a Themes page, which you can use to check yourselves. Further members of the reference group are still welcome. Please note that you may solve the problem sets in groups of up to four. For questions (regarding the contents of the course, or mathematics in general), my new office hours are: Tuesdays 13–14. Practical questions concerning the group sessions or the problem sets should in first line be posted to Karin. First problem set posted. Lecture notes online. Please note that the problem sets are mandatory, with deadlines for each set. First lecture: Thursday the 23th of August.

Time and place

Lectures will take place Thursdays 14.15 and Fridays 12.15 in lecture hall S1. There will be one problem session each week, starting the week after the first lecture. For details, see the schedule for TMA4145.

Lecturer

Homepage. My office is located at 1040, SB 2 (office hours: Tuesday 13–14 pm). Contact: mats [dot] ehrnstrom [at] ntnu [dot] no

Notes

• Th 22.11, Fr 23.11: Repetition.
• Th 15.11, Fr 16.11: Adjoints and decompositions. Self-adjoint operators and symmetric/Hermitian matrices. Unitary operators and unitary/orthogonal matrices. The spectral theorem for Hermitian/symmetric matrices. Positive definiteness. QR- (Gram–Schmidt), Cholesky- and singular value-decompositions. Suggested reading: Young: Sections 7.3 and 8.2. Friedberg: Sections 6.2–6.8, Strang: Sections 3.4, 5.2, 5.5, 6.1–6.3.
• Fr 9.11 Orthonormal sequences, Fourier coefficients and Fourer series. Bessel's inequality. Fourier coefficients are best possible (corollary: closest point). Convergence as an $l_2$-property. The Fourier series theorem. Suggested reading for this week: Young: Chapter 4 and Section 6.1 (Riesz representation theorem).
• Th 8.11 Orthogonal vectors and sets. The projection theorem (corollary: strict subspace characterization) and the Riesz representation theorem.
• Fr 2.11 Hilbert spaces. Convex sets and the closest point property. Suggested reading for this week: Young: Chapters 1 and 3.
• Th 1.11 Inner-product spaces and their properties. The Cauchy–Schwarz inequality, parallelogram law and polarization identity.
• Fr 26.10 Spectral decompositions: generalized eigenspaces. The Riesz index and a different characterization of algebraic multiplicity. Cayley–Hamilton and its consequences. The spectral decomposition, the Jordan form, and the spectral theorem for matrices. Suggested reading for this week (with discernment!): Friedberg: 5.1–5.2, 5.4, 6.6, 7.1–7.2.; Kreyszig: 7.1–7.2, Strang: Chapter 5 and Appendix B.
• Th 25.10 Constant-coefficient linear equations $\dot x = Ax$. Eigenvalues, eigenvectors and the spectrum of a bounded operator. Characteristic polynomials: the spectrum of a matrix consists of its eigenvalues. Algebraic and geometric mulitplicity of eigenvalues. Characterization of the solution space of $\dot x = Ax$. The exponential map for matrices and explicit solution formula.
• Fr 19.10 Contractions and the Banach fixed-point theorem; the Picard–Lindelöf theorem and Picard iteration. Suggested reading for this week: Kreyszig: Chapter 5.
• Th 18.10 Solving differential equations: initial-value problems; formulation as first-order systems; Peano's theorem; Lipschitz continuity.
• Fr 12.10 Equivalence of boundedness and continuity for linear operators. Finite-dimensional linear mappings are bounded. Kernels of continuous mappings are closed. Suggested reading for this week: Friedberg et al: 2.6; Kreyszig 2.10; Young: Chapters 6 and 7 (up to 7.2); Strang: 7.2 (the parts about the matrix norm).
• Th 11.10 : Boundedness. The operator norm and the normed space $B(X,Y)$. Functionals, duals, and the Riesz representation theorem. $B(X,Y)$ Banach for $Y$ Banach.
• Fr 5.10 : For matrices: null spaces, column spaces and row spaces; geometric interpretation of the nullspace. The rank–nullity theorem; reduction on $\mathbb R^n$, and geometric interpretation. Summary on linear equations (the Fredholm alternative). Suggested reading for this week: Friedberg et al: 2.1–2.4; Kreyszig 2.6 and 2.9 (not the part on functionals and duals); Strang 2.2, 2.4 and 2.6.
• Th 4.10 : The vector space $L(X,Y)$. Characterization of linear transformations on finite-dimensional vector spaces; matrix representations. Kernels and ranks. Characterization of injective linear transformations.
• Fr 28.9 : Gaussian elimination, LU- and LUP-decompositions. Gauss–Jordan elimination and the reduced row echelon form of a matrix. Linear transformations. Some important examples. Suggested reading: Friedberg et al: 2.1–2.2; Kreyszig 2.6; Strang 2.6.
• Th 27.9 : The change-of-basis matrix and its relation with invertible matrices. Linear systems, Gaussian elimination, and the row echelon form of matrix. LU- and LUP-decompositions. Gauss–Jordan elimination and the reduced row echelon form of a matrix. Suggested reading: Friedberg et al: 3.4 (especially 183–188); Strang 1.5, 2.2.
• Fr 21.9 : Characterization of finite-dimensional vector spaces. Schauder bases. Basis transformations: the change-of-basis matrix. Suggested reading: Friedberg et al: 2.5; Kreyszig 2.3; Strang 2.1, 2.3.
• Th 20.9 : Linear subspaces. Spaces of polynomials $P_n(R)$. Linear combinations, the span of a set, linear dependence. Hamel bases and the dimension of a vector space. Suggested reading: Friedberg et al: 1.3–1.6; Kreyszig 2.1 (2.4); Strang 2.1, 2.3.
• Fr 14.9 : Completions: isometries, isomorphisms, embeddings, and dense sets. Separability. The completion theorem. Suggested reading (sic!): Friedberg et al: 2.4 (on isomorphisms); Kreyszig 1.5–1.6; Young 21–22.
• Th 13.9 : Completeness proofs for $BC(I,\mathbb R)$ and $l_2$. Note on $BUC(I,\mathbb R)$. Suggested reading (sic!): Friedberg et al: 2.4 (on isomorphisms); Kreyszig 1.5–1.6; Young 21–22.
• Fr 7.9: Completeness: Cauchy sequences (convergent implies Cauchy, Cauchy sequences are bounded), complete metric spaces and Banach spaces, characterization of complete subsets. Suggested reading: Friedberg et al: 2.4 (on isomorphisms); Kreyszig 1.5–1.6; Young 21–22.
• Th 6.9: Examples of closed/open sets, closures. The unit ball is open. Continuous and sequential limits. Example: convergence in $BC(I,\mathbb R)$. Uniqueness of limits. Accumulation points. Relationship between limits, the distance function and closures. Suggested reading: Kreyszig 1.4.
• Fr 1.9: Metric spaces: discrete metric, Euclidean metric; normed spaces as metric spaces. Balls and spheres. $l_p$-spaces. The unit ball in different metrics. Interior points, boundary points, open and closed sets. Closures. The unit ball is open. Suggested reading: Kreyszig 1.1–1.3; Young Chapter 2 (the parts you understand).
• Th 30.8: Vector spaces: further examples. Normed spaces, $l_p$-norms, $BC$-spaces and equivalence of norms. Metric spaces. Suggested reading: Friedberg et al: 1.2; Kreyszig 2.1–2.2; Young 13–14; Strang 2.1.
• Fr 24.8: Unions, intersections and complements. Range of a function. Surjectivity, injectivity and invertibility. Countable/uncountable sets. Vector spaces: definition and first examples. Suggested reading: Friedberg et al: 6–12, Kreyszig 613–617, 50–53; Strang 69–71.
• Th 23.8:
• Introduction. Most important: i) always check the course web page (this page), ii) you need to hand in your solutions to the problem sets – in time.
• Theory. Sets and their notation. The sets $\mathbb N$, $\mathbb Z$, $\mathbb R$ and $\mathbb C$. Subsets and proper subsets ($C^k$-spaces). Cartesian products, (binary) relations, and functions. Difference between a function $f$ or $x \mapsto f(x)$ and its value $f(x)$ at the point $x$. Suggested reading: Kreyszig 609–617; Friedberg et al. 549–550.

Reference group

Recent feed-back

• The 'Themes' page, the 'Notes' and the lecture notes could benefit from more timely updates.
• In the lectures arguments sometimes need to be better explained, step by step.
• Before the exam it would be good with an overview of the entire course, accompanied by some words on what is important.
• The solutions to the problem sets need to be posted earlier, at least in good time for the exam.

If you feel that your thoughts on the course are not covered by the feed-back given by the reference group, please contact them, or me, in order to improve the course as much as is possible.

Exam

The exam is to take place on the 5th of December.