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# TMA4145 Linear Methods

• The exam with solutions has been posted (see below).

### Message board

 11.12 Finals 2013: bokmål nynorsk English Solutions

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### Time and place

Lectures are held Wednesdays 08.15 and Fridays 08.15 in S6. There will be one problem session each week, starting the week after the first lecture. For details, see the schedule for TMA4145.

Course contents and preliminary lecture plan

Course contents and preliminary lecture plan

 We. 21.08 Fr. 23.08 Introduction. Lecture notes sections 1.1–1.4 (Sets; Membership and inclusions; Set operations; Relations). Friedberg et al. 549–550; Kreyszig 609–617. Lecture notes sections 1.4–1.6. (Relations; Invertibility; Vector spaces) Friedberg et al: 6–12, Kreyszig 613–617, 50–53; Strang 69–71. Lecture notes sections 1.4–1.8. (Vector spaces; Normed spaces; Metric spaces) Friedberg et al: 1.2; Kreyszig 2.1–2.2; Young 13–14; Strang 2.1. Lecture notes sections 1.8–1.10. (Metric spaces; Balls and spheres; Interior points, boundary points, open and closed sets.) Kreyszig 1.1–1.3; Young Chapter 2 (the parts you understand). Lecture notes sections 1.10–1.11. (Interior points, boundary points, open and closed sets; Limits) Kreyszig 1.4. Lecture notes sections 1.11–1.12. (Limits; Completeness) Friedberg et al: 2.4 (on isomorphisms); Kreyszig 1.5–1.6; Young 21–22. Lecture notes section 1.12 (Completeness) Friedberg et al: 2.4 (on isomorphisms); Kreyszig 1.5–1.6; Young 21–22. Lecture notes section 1.13 (Completions) Friedberg et al: 2.4 (on isomorphisms); Kreyszig 1.5–1.6; Young 21–22. Lecture notes sections 2.1–2.3. (Linear subspaces; linear dependence; Bases and dimension) Friedberg et al: 1.3–1.6; Kreyszig 2.1 (2.4); Strang 2.1, 2.3. Lecture notes sections 2.3–2.5. (Bases and dimension; Schauder bases; Basis transformations) Friedberg et al: 2.5; Kreyszig 2.3; Strang 2.1, 2.3. Lecture notes sections 2.5–2.6 (Basis transformations; Gaussian elimination) Friedberg et al: 3.4 (especially 183–188); Strang 1.5, 2.2. Lecture notes sections 2.6–2.7 (Gaussian elimination; Linear transformations) Friedberg et al: 2.1–2.2; Kreyszig 2.6; Strang 2.6. Lecture notes section 2.7 (Linear transformations): 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. 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. Lecture notes section 2.7 (Linear transformations): 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). Lecture notes section 2.8 (Bounded linear transformations): 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. 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). Lecture notes section 2.8 (Bounded linear transformations): Equivalence of boundedness and continuity for linear operators. Finite-dimensional linear mappings are bounded. Kernels of continuous mappings are closed. Solving differential equations: initial-value problems; formulation as first-order systems; Peano's theorem; Lipschitz continuity. Kreyszig: Chapter 5. Contractions and the Banach fixed-point theorem; the Picard–Lindelöf theorem and Picard iteration. 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. 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. 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. Inner-product spaces and their properties. The Cauchy–Schwarz inequality, parallelogram law and polarization identity. Young: Chapters 1 and 3. Hilbert spaces. Convex sets and the closest point property. Orthogonal vectors and sets. The projection theorem (corollary: strict subspace characterization) and the Riesz representation theorem. Young: Chapter 4 and Section 6.1 (Riesz representation theorem). 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. 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. 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. Repetition.

### Lecturer

Homepage. My office is located at 1146, SBII. Visit me in my office hours Tuesdays 11–12 pm (preferred), or contact me at mats [dot] ehrnstrom [at] ntnu [dot] no

### Reference group

Recent feedback:

• Solutions to the exam August 2013 was asked for. It is not probable that these will be written. The advice is to focus on those problems from that exam that have been given as part of the problem sets. Also don't forget the exam page.
• The recent pace (Hilbert spaces) might be too high (though there were differing reports on this).
• Parts of the reference group report that they currently have less time for the course, but that, in general, matters seem to be fine.

Here is the report from last years reference group.

### Exam

The exam is to take place on the 11th of December. Have a look at last year's problems and statistics already now.