1S2: Discrete Mathematics I for Scientists
Notes
Some (not all) parts of the course notes will be in the form of a handout or will be available here. All the are in PDF format and require a programme such as Adobe Acrobat Reader to read them.
- Chapter 0. What is Linear Algebra?
- Some short introductory remarks.
- Chapter 1: Linear Equations
- These notes deal with Gaussian elimination and Gauss-Jordan elimination, as ways of solving systems of linear equations.
- Chapter 2: Simple UNIX comands and spreadsheets
- These notes are in two parts. The first part offers a few basic ideas about how to use the Maths (UNIX - FreeBSD and Linux) computer system while the second part has some very basic ideas about what spreadsheets are good for.
- Chapter 3: Vectors
- These notes deal with vectors from a geometrical point of view (arrows) first. Then a more algebraic approach (with components). Equations of lines and planes in space. Basic ideas about higher dimensions.
- Chapter 4: Mathematica
- This is a very quick introduction to how to use the computer programme Mathematica. At the end there are some things about matrices, some of which use concepts we have yet to encounter.
- Chapter 5: Matrices
- These notes deal with matrix operations (addition, muliplication by scalars, matrix multiplication). They continue with expressing elementary row operations via matrix multiplication by elementary matrices, inverses, how to find inverses. Next special kinds of square matrices (diagonal matrices, upper triangular matrices, strictly upper triangular, nilpotent matrices, lower triangular). Transposes. Traces of (square) matrices. (Still to be checked.)
- Chapter 6: Determinants
- Derminants of square matrices; evaluation by cofactor expansion and by row reduction; link with invertiblity; main properties of determinants. Determinants give areas and volumes. Cross products. Use of determinants to find equations. Vertex matrices. (Still to be checked.)
- Chapter 7: Binary, octal and hexadecimal numbers
- These notes deal with elementary matters about base 2, base 8 and base 16, how these relate to one another and to the way computers store numbers (integer and floating point storage). The limitations on size and accuracy of the standard methods computer use are considered.
- Chapter 8: Linear trsnsformations
- We present the idea of a matrix giving rise to a linear transformation, and some concrete examples (diagonal matrices, rotations). Orthogonal matrices. 3 by 3 rotations, orthononormal bases. Linear transformations in the abtract. Eigenvalues, orthogonal diagonalisation. Diagonalisation and application to differential equations. Least squares fit.
- Chapter 9: An Introduction to Probability and Statistics
- We introduce some basic concepts of probability (sample space, event, probability, random variable, mean, variance), a very little on data and some of the examples of probability distributions most used in applications (binomial, Poisson and normal). In the normal case we are dealing with a continuous distribution and improper integrals enter in.