MATH1061 Notes (1st in course)

Description

Notes summarising the content of Mathematics 1A (MATH1061) with additional content from advanced lectures. Week 1: Motivation, vector basics, R^n, vector properties, vector spaces. Week 2: Vector length, unit vectors, dot product, dot product properties, Cauchy-Schwarz inequality, triangle inequality, projections, cross product. Week 3: Cross product, scalar triple product, lines, planes, distances between points, lines, and planes. Week 4: Systems of linear equations, augmented matrices, Gaussian elimination, Gauss-Jordan elimination, row echelon form, reduced row echelon form, consistency. Week 5: Matrix notation, matrix algebra, matrix multiplication, transpose, trace, inverse matrices. Week 6: Computing inverses, elementary matrices, characterisations of invertibility, span, linear independence, linear dependence. Week 7: Subspaces, spanning sets, bases, dimension, row space, column space, null space. Week 8: Bases for row, column, and null spaces using RREF, rank, nullity, rank-nullity theorem, further invertibility characterisations. Week 9: Markov chains, probability vectors, stochastic matrices, transition matrices, steady-state vectors, introduction to eigenvalues and eigenvectors. Week 10: Determinants, minors, cofactors, Laplace expansion, determinant row-operation rules, adjugate formula, eigenvalues, eigenvectors, eigenspaces, algebraic multiplicity, geometric multiplicity. Week 11: Similar matrices, invariants under similarity, diagonalisation, eigenbasis criterion. Week 12: More Markov chains, regular stochastic matrices, convergence to steady state, population growth, Leslie matrices, Google PageRank. Week 13: Summary and exam preparation, definition checklist, calculation templates, common exam-style linear algebra tasks.