MA1S12: JF Mathematics 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 will be in PDF format and
require a programme such as
Adobe Acrobat Reader to read them.
Chapter 1, Determinants
Definition by cofactor expansion along the first row;
properties;
evaluation by row operatiors,
using cross products and scalar triple products to get a geometrical
interpretation of 2 by 2 and 3 by 3 determinants as areas/volumes,
the adjugate matrix and a formula for the inverse matrix,
Cramer's rule and some uses of derminants to find equations of
lines, circles and planes.
Chapter 2, Linear transformations
Matrices viewed as giving linear transformations, geometrical
examples in 2 and 3 dimensions of linear maps, including
rotations in 2 and
3 dimensions. Othogonal matrices. Change of basis to a new orthonormal
basis.
Gram-Schnidt procedure (in 3 dimensions).
Finding the matrix for a rotation given the axis and the angle.
Finding the axis of rotation and the angle from the matrix.
The abstract definition of a linear transformation.
Chapter 3, Eigenvalues, diagonalisation and some
applications
Eigenvalues, orthogonal diagonalisation.
Diagonalisation. Matrix exponentials. (Systsems of) Linear ordinary
differential equations.
Least squares. Markov matrices.
Chapter 4, An Introduction to Probability and Statistics
Sample space, probability, event, random variable, (theoretical)
mean and variance and standard deviation. Sample mean and variance.
Conditional probbaility.
Binomial distribution(s).
Poisson distribution.
Continuous distributions especially the normal distribution(s) (and the probability density and probability distribution functions).
Notion of a confidence interval and use of the Student $t$-tables.
