TMA4110 Matematikk 3 Fall 2013, BYGG, ING, PETR; TEKGEO

• Tuesday 8.15-10.00 (R1)
• Thursday 14.15-16.00 (F1)

• Monday 14-15 (rom 948, SB II)

• Information about exercise 13 are now given under "Homeworks".

• We have the reference group, click the meny on the left to see the names and e-mail addresses. The first meeting is on Friday, September 6th.

• Lecture Notes are below.

• The lectures for this parallel will be given in English. First lecture is on Tuesday, August 20th, 8.15 in R1. Welcome!

20.08 Lecture 1
We discussed real numbers, using number line, and defined complex numbers as ordered pairs of reals; each complex number corresponds to a point on the plane. This gives representations of complex numbers in Cartesian and polar coordinates. Arithmetic operations on complex numbers were introduced and a number of examples computed. We learned what are real and imaginary parts of a complex number, defined the absolute value and complex conjugate of a complex number.
22.08 Lecture 2
First, we repeated the main concepts form the first lecture and went through some examples. We concentrated on polar form of complex numbers, computed the absolute value and argument of the product of two complex numbers; discussed powers of complex numbers (De Moivre's formulas), started with the roots of complex numbers.
27.08 Lecture 3
We discussed roots of unity and did an example with cube roots of a complex number (Problem 1 in August 2012 exam). Then we used formula for the roots of quadratic polynomial to find complex roots, we learned also that if coefficients of a polynomial are real then the complex conjugate of a root is also a root. We discussed the fundamental theorem of algebra and started complex exponential function.
29.08 Lecture 4
Exponential complex function and complex-valued functions were discussed. We found the derivative of the function f(t)=exp((a+ib)t). Second order linear differential eqauations were introduced and mass-spting system discussed. We defined linearly independent functions and Wronskian of two functions. Initial value problem was discussed by an example.
03.09 Lecture 5
We continued the discussion of second order linear homogeneous differential equations; looked again at the Wronskian of two solutions and derived the Abel formula. Finally, we found general solutions of homogeneous equations with constant coefficients and computed a number of examples.
05.09 Lecture 6
Mass-spring system with damping was studied first. We started with harmonic oscillation (no damping) and saw how the solution chenges when we increase the damping constant. Three cases can be distinguished: underdamping, critical damping, and over damping. We used symbolic solution of ODE from MATLAB ('dsolve') to copmute the solutions and plotted them again using MATLAB. Then the method of undetermined coefficients was discussed.
10.09 Lecture 7
We worked on an example of the method of undetermined coefficients first. Then we discussed forced oscillation with periodic external force. The resonance for equation without damping. Equation with damping: transient solution, steady-state solution and gain.
12.09 Lecture 8 part 1 part 2
The lecture consisted of two parts: Variation of parameters for inhomogeneous second order ODE and introduction to linear algebra
17.09 Lecture 9
We discussed the Gauss elimination, vectors in n-dimensional space and operations on them, matrix equation Ax=b.
19.09 Lecture 10
Matrix-vector multiplication was discussed further. Solutions of homogeneous and inhomogeneous systems were compared and some examples given; we learned also that each homogeneous system with more variables than equations has infinitely many solutions. Then three applications of linear systems were given following the topics in Section 1.6 of the textbook.
24.09 Lecture 11
We discussed linear (in)dependent sets of vectors and gave a number of examples. At the second hour linear transformations were introduced with examples of geometric transformations on the plane.
26.09 Lecture 12
We continued to discuss matrices of lienear transformations and computed the standard matrix of a rotation on the plane. Further, we defined transformations that are onto and one-to-one and discussed how one can deduce this properties looking at the matrix. Two old exam problems were considered. Finally, we briefly discussed the population model from Section 1.10.
01.10 Lecture 13
We started matrix operations and introduced matrix multiplication by considering compositions of linear transfromations. A number of examples were given. The transpose matrix was defined and the transpose of the product computed.
03.10 Lecture 14
We defined the inverse matrix and gave an algorithm that computes the inverse when it exists. We also looked at invertability of the standard matrix of a linear transformation from n-dimensional space to itself.
08.10 Lecture 15
We defined determinants of square matrices and formulated theorem on cofactor expansion. Then we demonstrated how determinants can be calculated using elementary row transformations.
10.10 Lecture 16
We finished the third chapter, did some problems on determinants, discussed Cramers rules, calculation of the inverse matrix, using cofactors and briefly mentioned connections between determinants and areas and volums. Next time we start chapter 4, 4.1 is a section on vector spaces and subspaces, read it in advance.
15.10 Lecture 17
We defined subspaces of Rⁿ and gave a number of examples, including spans of vectors and solution sets of linear systems. Then we introduced abstract vector spaces.
17.10 Lecture 18 We discussed the column and solution spaces of a matrix, including the connection of these spaces to the corresponding linear transformation
22-24.10 Lectures 19-20 The topic this week was: bases for vector spaces, dimension, rank of a matrix. We also looked at coordinat systems and coordinates related to a basis (4.4) and Markov chains (4.9).
29.10 Lectures 21 We started a new chapter (Ch. 5) and new topic, eigenvector and eigenvalues of square matrices and discussed a number of examples. The charectiristic polynomial of a matrix was defined, the zeros of the characteristic polynomial are the eigenvalues.
31.10 Lectures 22 The topic was diagonalization of matrices. To check that an n×n matrix is diagonalizable one has to check that it has n linearly independent eigenvectors.
5-7.11 Lectures 23-24 This week we studied complex eigenvectors and eigenvalues of matrices with real elements, specially two-by-two matrices. Then we applied the eigenvaøues and eigenvectors computations to linear systems of first order differential equations with constant coefficients.
12.11 Lectures 25 We started orthogonal projections and orthogonal complements. It helps to have two- or three-dimensional pictures in mind when studing more general theory.
14.11 Lectures 26 We discussed formulas for orthogonal projection (do not forget to check that you have an orthogonal basis before you apply it) and Gram-Schmitd algorithm.
29.10 Lectures 27 Todays main topic is the least squares solution and line fitting. We do some examples and exam problems. We also start symmetric matrices and orthogonal diagonalization.
29.10 Lectures 28 The last topic of the course is quadratic forms and conic sections.