Matrix Algebra
This is a minicourse in linear algebra. We follow a geometric approach, which allows one to produce the shortest proofs possible. The theory is presented as a series of exercises, and each exercise typically contains no more than one idea. Each section has a summary with questions for repetition.
Clarity of ideas is put above strictly logical sequencing. Omission of boring details is indicated explicitly. Complex results are given where the reader is thought to be prepared for them. One notable example is determinants, which usually are given in the beginning of a linear algebra course but here are placed closer to the end. In short, this is the course I would prefer when I was a bachelor student.
The course completely covers the theory required for MT2175 "Further Linear Algebra"(and at a higher level than Anthony and Harvey, Linear algebra: Concepts and methods).
Section 1. Matrix operations
Matrix notation and summation, columns and row vectors, axioms of summation
Vector and matrix multiplication, scalar product and its symmetry, partitioning and compatibility rule
Roadmap for studying matrix multiplication, axioms of multiplication, commutativity, associativity and distributivity
Matrix inversion: doing some housekeeping at elementary level, identity matrix, invertibility, inverse of an inverse and inverse of a product
From invertibility to determinants: argument is more important than result, determinant, multiplicativity, Invertibility condition, determinant of an inverse, right and left inverses
Matrix transposition: continuing learning by doing, symmetric matrix, determinant of a transpose, transpose of a product, transpose of an inverse
Matrix algebra: questions for repetition
Section 2. Vector operations, scalar product and related
Euclidean space geometry: vector operations, parallelogram rule, unit vector, scaling, linear combination
Euclidean space geometry: scalar product, norm and distance, homogeneity, additivity, orthogonal vectors, arithmetic square root, Pythagoras theorem
Euclidean space geometry: Cauchy-Schwarz inequality, triangle inequality, cosine of the angle between vectors, distance, ball, open set
Euclidean space geometry: questions for repetition, scalar product, norm and distance properties; parallelogram law
Section 3. Geometry of linear equations
Geometry of linear equations: matrix as a mapping, image, counter-image, first characterization of matrix image, multiplication rule for partitioned matrices
Geometry of linear equations: linear spaces and subspaces, subspace spanned by vectors
Geometry of linear equations: structure of image and null space, homogeneous and inhomogeneous equations, hyperplane
Geometry of linear equations: orthogonal complement and equation solvability, second orthocomplement, second characterization of matrix image
Geometry of linear equations: questions for repetition
Section 4. Equation solvability and linear (in)dependence
Is the inverse of a linear mapping linear? Orthonormal system and basis, matrix generated by linear mapping
Solvability of an equation with a square matrix: link between invertibility, null space, image and determinant
Linear dependence of vectors: definition and principal result, trivial cases, case of two vectors, criterion of linear independence
Basis and dimension, intersection of orthogonal subspaces, dimension additivity
Correctness of the space dimension definition
Rank of a matrix and the rank-nullity theorem, rank of a matrix and its transpose, upper bound for rank
Final touches on linear independence One more definition, correctness of the dimension definition, linear dependence of a large number of vectors
Summary and questions for repetition
Section 5. Spectral theory and diagonalization by orthogonal matrices
Complex numbers: time to turn on the beacon Imaginary unit, conjugate, absolute value, polar form
Law and order in the set of matrices Motivation, definition and geometry of order
Matrix similarity Changing bases, motivation for matrix similarity
Orthogonal matrices Algebra, geometry and link to the transition matrix
Eigenvalues and eigenvectors Motivation, link to characteristic equation, fundamental theorem of algebra
General properties of symmetric matrices The role of 
Diagonalization of symmetric matrices Main result: symmetry is necessary and sufficient for diagonalizability
Questions for repetition: diagonalization by orthogonal matrices
Section 6. Applications of the diagonalization theorem
Applications of the diagonal representation I Matrix positivity, matrix functions and linear differential equation
Applications of the diagonal representation II Square root of a matrix and Generalized Least Squares estimator
Applications of the diagonal representation III Absolute value and polar form for matrices
Applications of the diagonal representation IV Principal component analysis
Questions for repetition Representing a complex matrix as a linear combination of symmetric matrices and an exercise on elementary matrices.
Section 7. Projectors and their applications
Geometry and algebra of projectors Includes the characterization of the image and null space of a projector
Constructing a projector onto a given subspace The definition requires a basis of the subspace.
Application: Ordinary Least Squares estimator Includes a version of the Pythagoras theorem
Eigenvalues and eigenvectors of a projector Includes trace of a projector
Application: estimating sigma squared This is about estimating the error variance in multiple regression
Application: distribution of the estimator of the error variance
Distributions derived from normal variables Joint density of independent variables, standard normal and general normal variables
Application: distribution of sigma squared estimator More on projectors related to OLS
Questions for repetition: projectors and applications
Section 8. Determinants
Axioms 1-3 and Properties I-III If rows of a matrix are linearly dependent, its determinant is zero
Properties IV-VI Multilinearity and antisymmetry
Permutation matrices Definition and example
Properties of permutation matrices Pre-multiplication by permutation matrix and orthogonality
Leibniz formula for determinants Just an application of multilinearity
Determinant of a product Another illustration of multilinearity
Determinant of a transpose One more use of multilinearity
Cramer's rule and invertibility criterion Short and elegant proofs
Laplace expansion With special case to show intuition
Determinants: questions for repetition
Section 9. The Jordan form: geometry
Direct sums of subspaces Oblique projector, dimension additivity
Properties of root subspaces Inclusion, stabilization, related space decomposition into a direct sum
Action of a matrix in its root subspace Root vectors, their linear independence, Jordan cell
Chipping off root subspaces The whole space is a direct sum of root subspaces
Playing with bases Basis existence, relative linear independence and relative basis
Main theorem: Jordan normal form
Section 10. Sign-definite matrices
Elementary transformations Links between geometric and matrix definitions
Gaussian elimination method and its theoretical treatment
Sylvester's criterion for positive definite and negative definite matrices
