UNIVERSITY OF DUBLIN
Course 1S2
TRINITY COLLEGE
JF Mathematics
Natural Sciences
Two lectures for course 1S2 are given each week with regular tutorials during term.
Tutorial assignments contribute at most 1% for each tutorial to the overall 1S2 grade.
A 3 hour annual examination for course 1S2 is held at the end of the academic year.
The course presents vector methods, linear systems of equations, an introduction to
matrix algebra, and ordinary differential equations important for the natural sciences.
The following summarizes the areas covered including applications to problems in the
physical, chemical, earth and biological sciences.
• Vectors; vector addition, norm of a vector, dot product of two vectors, angle
between two vectors with applications (Anton and Busby: Chapter 1).
• Systems of linear equations; row reduction, Gauss-Jordan elimination, solution
set for equations, applications (Anton and Busby: Chapter 2).
• Matrices; matrix algebra, inverse and transpose of a matrix; unit, diagonal and
symmetric matrices; trace of a matrix (Anton and Busby: Chapter 3).
• Determinants; evaluation by row operations, cross product, determinants and
matrix eigenvalues and eigenvectors (Anton and Busby: Chapter 4).
• Matrix Models; dynamical systems and Markov matrices, applications to
geographical population movements (Anton and Busby: Chapter 5).
• Differential equations of first and second order ; applications to predator-prey
and symbiotic dynamics (Anton and Busby: Section 8.10).
Textbooks:
1. H. Anton & R. C. Busby, Contemporary Linear Algebra, John Wiley & Sons,
Obtain this textbook, or the next textbook if preferred.
2. H. Anton and C. Rorres, Elementary Linear Algebra, (applications version, if
preferred), (www.wiley.com/college/anton).
3. David C. Lay, Linear Algebra and its applications, (www.laylinalgebra.com),
Addison-Wesley Longman.
4. Ron Larson and B. Edwards, Elementary Linear Algebra, (http:// college.
hmco.com / mathematics / larson / elementary linear / 4e / students /),
Houghton Mifflin Company.
5. B. Kolman and D. R. Hill, Introductory (or Elementary) Linear Algebra with
applications, Prentice Hall.
Course 1S2 Web Page
http://www.maths.tcd.ie/~nhb/1S2.php
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Annual mathematics examination papers over recent years are available from the Local
Home Page of www.tcd.ie/ by selecting (under For Students) the links
1. Examination Papers
2. Annual Examinations
3. Science (Course) Junior Freshman (Standing) Year (Academic Year)
4. MA1S21 (Maths 1S2), or MA1S41 (Maths 1S4).
Course 1S2 (formerly 1S4) is based upon either of the first two textbooks listed above.
The material is similar though the order of topics is rearranged. Other textbooks may
suit students with particular backgrounds or ways of understanding mathematics.
The beginnings of matrices go back to the second century BC when the Chinese
considered problems like the following, taken from matrices and determinants in
www-history.mcs.st-and.ac.uk/history/Indexes/Algebra.html
There are three types of corn, of which three bundles of the first, two of
the second, and one of the third make 39 measures. Two of the first, three
of the second and one of the third make 34 measures. And one of the first,
two of the second and three of the third make 26 measures. How many
measures of corn are contained of one bundle of each type?
Suppose x denotes the measure of corn contained in one bundle of the first type, y
denotes the measure of corn contained in one bundle of the second type, and z denotes
the measure of corn contained in one bundle of the third type. The system of linear
equations in x, y and z is given by
3x + 2y + z = 39
2x + 3y + z = 34
x + 2y + 3z = 26 .
Course 1S2 will introduce elementary row operations and the Gauss–Jordan method of
finding the solution set of such a system of linear equations. Another important topic
relates to the determinant of a matrix.
In 1683 Takakazu Seki, a prodigy, published the idea of a determinant in Japan.
The word ‘determinant’ was first introduced over two hundred years ago by Gauss in
Disquisitiones arithmeticæ (1801). Karl Friedrich Gauss (1777–1855) was a German
mathematician and scientist who enjoyed mineralogy and botany as hobbies.
Wilhelm Jordan (1842–1899) was a German engineer whose method for solving
linear systems appeared in Handbuch der Vermessungskunde (Handbook of Geodesy)
in 1888.
The 3-hour examination for course 1S2 at the end of the academic year will involve
attempting six questions from a total of eight questions.
Dr. Buttimore
nhb @ maths.tcd.ie
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School of Mathematics