TMA4145 Linear Methods

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.

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

• 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.

• Merete Falck [MTKJ]
• Per Kristian Ørke [BMAT]
• Thomas Øvestad [MTFYMA]
• Olav Willumsen Haugå [MAFYMA]

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.

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