# Bridging Sessions: KS5 to Uni

#### AS Mathematics Flipped Videos

FMSP Wales is running an exciting series of pre-recorded presentations which will enable year 13 students to bridge towards studying mathematics at university. There is rarely time during the school teaching framework to see how differential equations can be used to model epidemics; or to investigate how complex numbers generate the amazing Mandel brot set. The present course is an opportunity to linkup with Higher Education to benefit from their deep knowledge of the subject and their intimate familiarity with university-level mathematics. The programme is recommended to any students considering taking a maths-rich degree in September 2020. Each session will be pre-recorded and approximately 30minutes long. They will be released weekly from Wednesday 6thMay 2020, and each made available for 2 weeks. There will be an opportunity for students to ask the lecturer questions about the presentation by email, and the lecturer will release a video response to selected questions, in the second week

##### 6 May 2020: “Eigenvectors, Eigenvalues and the Cayley-Hamilton theorem” Adrian Wells, FMSP Wales

Eigen is a German word with several meanings, more usually considered to be particular or peculiar (which does make it interesting as a mathematical concept). In short, an eigen vector is a vector which is mapped by some operation on to a multiple of itself (this multiple is described as the related eigenvalue). Typically, the operation utilised is multiplication by a square matrix, since these can be used to represent transformations in 2 and 3 dimensions. Study and evaluation of eigenvalues and vectors gives rise to a characteristic equation directly linked to the matrix being used. This equation gives direct rise to the Cayley-Hamilton theorem: Any matrix will satisfy its own characteristic equation, perhaps a little simplified but it then allows an alternative approach to evaluating powers of the matrix. Not too much of a problem with 2×2 matrices but progressively more challenging as the square matrix becomes bigger in dimension. Prior knowledge required: It is anticipated that not all participants will have studied further maths but a brief introduction to 2×2 matrix algebra: determinants, multiplication and transformations (think GCSE, reflection, rotation and enlargement) would enable you to gain more from the session. A brief work through these topics can be found on the Khan Academy website link, if you are able to access it the section on matrices and transformation to be found on Integral is particularly good link. There are, of course, several other articles and videos to be found on the web.

Useful for undergraduate modules: Linear Algebra, Vector Spaces, Methods of Algebra and Calculus