## Lecturer Dmitri Zaitsev

This course corresponds to the 2nd half of the course 2E1 of the last year 2006-07.

Problem Sheets in PDF: Sheet 1 Sheet 2 Sheet 3 Sheet 4 Sheet 5 Sheet 6 Sheet 7 Sheet 8 Sheet 9 Sheet 10 Sheet 11 Sheet 12 Sheet 13

Most solutions to the exercises are very similar to those older ones that can be found here, here and here. Here solutions of the few excercises that are slightly different: Sheet 12

Every problem in the calculus of variations has a solution, provided the word solution is suitably understood. -- David Hilbert

Course outline:

Linear Algebra (Chapters 4-7 in Anton-Rorres' book). Euclidean n-Space and n-Vectors, Operations with them. Linear Transformations and their Matrices. Subspaces. Linear Combinations of Vectors. Subspaces spanned by a Set of Vectors. Linear Independence of a Set of Vectors. Basis and Dimension. Standard Basis in n-space. Coordinates of Vectors relative to a Basis. General and Particular Solutions for a Linear System. Row, Column and Nullspace of a Matrix. Finding Bases for them using Elementary Row Operations. Rank and Nullity of a Matrix. Inner Products, Lengths, Distances and Angles relative to them. Orthogonal and Orthonormal Bases relative to an Inner Product. Orthogonal projections to Subspaces. Gram-Schmidt Process (see Example 7 in Chapter 6.3). Best Approximation by the Least Squares method. Eigenvalues and Eigenvectors of Square Matrices.

Fourier Analysis (Chapter 10 in Kreyszig' book). Fourier Series for periodic functions. Euler Formulas for the Fourier Coefficients. Even and Odd Functions. Fourier Cosine and Fourier Sine Series for them. Fourier Integral and Fourier Transform.

Calculus:
Calculus for Beginners and Artists by Daniel Kleitman.
Multivariable Calculus Online by Jeff Knisley
Taylor's formula, linear and quadratic approximations by Eric A. Carlen
Linear Algebra:
Working with Vectors (A Self-Help Workbook for Science and Engineering Students by Jenny Olive)
Java applet introducing 3-vectors by Maths Online
Matrix Algebra Tutorials by S.O.S. MATHematics
A Linear Algebra book by Jim Hefferon (PDF file)
Importance of Linear algebra in Engineering Design Methodology by Mysore Narayanan (PDF file)

Old 2E1 web pages.
2E1 2006-07 Part I by Richard Timoney and Part II by Dmitri Zaitsev with Problem Sheets and some Solutions.
2E1 2005-06 by Dmitri Zaitsev with Problem Sheets and some Solutions.
2E1 2004-05 by Dmitri Zaitsev with Problem Sheets and some Solutions.
2E1 2003-04 by Fermin Viniegra with many interesting links.

For exam-related problems look in TCD past examination papers and Mathematics department examination papers.

I will appreciate any (also critical) suggestions that you may have for the next term. Let me know your opinion, what can/should be improved, avoided etc. and I will do my best to follow them. Please use the feedback form from where you can also send anonymous messages.