| Sheet | PS | |
| Series terminology | download | download |
| Matrices terminology | download | download |
| Week | Lecture | Date | Topic | In Riley et al at... |
| 11 | 1 2 3 |
|
Collections exam Series basics Tricks for summing series |
4.1,4.2 4.2 |
| 12 | 4 5 6 |
Convergence of infinite series; comparison and quotient tests. Ratio, Cauchy's, and integral tests. Convergence for alternating series, absolute and conditional convergence. Power series. |
4.3 4.3 4.3,4.5 | |
| 13 | 7 8 9 |
Convergence of power series, interval of convergence. Operations on power series Taylor polynomials, examples. Taylor series, Lagrange form of the remainder, approximation error, Taylor's theorem |
4.5 4.6 4.6 |
|
| 14 | 10 11 12 |
Taylor series and limits Matrices: motivation, simple matrix operations, matrix multiplication Matrix transpose, functions of matrices |
4.6 8.3,8.4 8.6,8.5 |
|
| 15 | 13 14 15 |
Systems of linear equations: examples Systems of linear equations: Gaussian elimination, row echelon form Homogeneous systems. Vector spaces, linear combinations of vectors |
8.18 8.18 8.18,8.1 |
|
| 16 | 16 17 18 |
Vector spaces: linear independence, basis Linear maps, linear operators, and their matrices Kernel, image, rank of a linear map |
8.1 8.2 8.2, 8.11 |
|
| 17 | 19 20 21 |
Rank of matrix and elementary row operations. Trace and determinant Calculation of determinant, properties, determinants and Gauss elimination Determinant and rank. Inverse matrix using elementary row operations |
8.18, 8.8, 8.9 8.9 8.11,8.10 |
|
| 18 | 22 23 24 |
Inverse matrix using cofactors Symmetric, orthogonal, Hermitian, unitary matrices Change of basis |
8.10 8.7,8.12 8.15 |
|
| 19 | 25 26 27 |
Eigenvalues and eigenvectors: definition, characteristic equation and polynomial Determining eigenvectors, examples. Properties of eigenvalues and eigenvectors Eigenvalues of Hermitian matrices. Diagonalisation: algebraic and geometric multiplicities |
8.13,8.14 8.14 8.13,8.16 |
|
| 20 | 28 29 30 |
Diagonalisation of matrices: algorithm, finding a transformation matrix Diagonalisation of matrices: examples. Polynomial and exponent of a matrix Diagonalisation of Hermitian matrices. Cayley-Hamilton Theorem |
8.16 8.16 8.16 |
|
| 21 | 31 32 33 |
Quadratic forms Diagonalisation of quadratic forms Revision |
8.17 8.17 - |