Author(s): N. Peake, ed. Dexter Chua
Series: Cambridge Mathematical Tripos Part IA Lecture Notes
Publisher: University of Cambridge
Year: 2014
Language: English
City: Cambridge
Tags: maths; mathematics; math; advanced; college; university; higher; further; pure; applied; linear algebra
Introduction
Complex numbers
Basic properties
Complex exponential function
Roots of unity
Complex logarithm and power
De Moivre's theorem
Lines and circles in C
Vectors
Definition and basic properties
Scalar product
Geometric picture (R2 and R3 only)
General algebraic definition
Cauchy-Schwarz inequality
Vector product
Scalar triple product
Spanning sets and bases
2D space
3D space
Rn space
Cn space
Vector subspaces
Suffix notation
Geometry
Lines
Plane
Vector equations
Linear maps
Examples
Rotation in R3
Reflection in R3
Linear Maps
Rank and nullity
Matrices
Examples
Matrix Algebra
Decomposition of an n x n matrix
Matrix inverse
Determinants
Permutations
Properties of determinants
Minors and Cofactors
Matrices and linear equations
Simple example, 2 x 2
Inverse of an n x n matrix
Homogeneous and inhomogeneous equations
Gaussian elimination
Matrix rank
Homogeneous problem Ax = 0
Geometrical interpretation
Linear mapping view of Ax = 0
General solution of Ax = d
Eigenvalues and eigenvectors
Preliminaries and definitions
Linearly independent eigenvectors
Transformation matrices
Transformation law for vectors
Transformation law for matrix
Similar matrices
Diagonalizable matrices
Canonical (Jordan normal) form
Cayley-Hamilton Theorem
Eigenvalues and eigenvectors of a Hermitian matrix
Eigenvalues and eigenvectors
Gram-Schmidt orthogonalization (non-examinable)
Unitary transformation
Diagonalization of n x n Hermitian matrices
Normal matrices
Quadratic forms and conics
Quadrics and conics
Quadrics
Conic sections (n = 2)
Focus-directrix property
Transformation groups
Groups of orthogonal matrices
Length preserving matrices
Lorentz transformations
