university levels of education Linear Algebra I — V
Five levels in one reader, from a pair of numbers on a grid to the spectral theorem, Jordan form and PageRank — handmade, no ads, no sign-ups. A hundred and fifteen written lectures with matrices and vectors properly typeset, three hundred and sixty-five worked examples solved step by step with the reasoning shown, four hundred and sixty practice questions with hints and answers, quizzes for every lecture, and two hundred and fifty unsolved problems to do on your own, in nineteen ways of studying including a continuous read of a whole course. Start at whichever level you need; the last entry opens all five courses as one book.
- 001
Linear Algebra I
Vectors, Systems and First Matrices. The objects of the subject and the arithmetic that goes with them — vectors read both as lists and as arrows, addition and scaling, linear combinations, the vector between two points, magnitude and unit vectors, and the move into three dimensions. Then the dot product and the geometry it encodes: angles, orthogonality, projection and the cross product. Systems of equations solved by substitution and elimination, augmented matrices and row operations, Gaussian elimination and back substitution, how to tell one solution from none from infinitely many, and parametric answers. Closing with matrix arithmetic: addition, the transpose, the matrix–vector product read as a combination of columns, matrix multiplication, rectangular shapes, and the trace. Twenty-six lectures, 312 questions. Every lecture carries a written explanation with the mathematics properly typeset — matrices, vectors and Greek letters rendered as real notation — a table of the formulas that matter, defined terms and the mistakes to watch for, worked examples revealed step by step with the reasoning beside each step, practice questions with hints, a quiz, and unsolved problems to do on your own, studied through nineteen modes, from flashcards and drills to timed exams and a continuous read of the whole course as one page.
live - 002
Linear Algebra II
Matrix Algebra, Determinants and Vector Spaces. Which laws of ordinary algebra survive the move to matrices and which quietly fail — powers, the reversal rules, commutativity, elementary matrices, matrix equations, symmetry and skew-symmetry. Then determinants: the 2×2 formula, cofactor expansion, minors, triangular shortcuts, how each row operation changes the answer, the singular-or-invertible test and Cramer's rule. Inverses by formula, by Gauss–Jordan and by adjugate. Finally the abstract core — subspace tests, span, linear independence, bases and dimension, column space, null space, rank–nullity and coordinates in a chosen basis. Twenty-seven lectures, 324 questions. Every lecture carries a written explanation with the mathematics properly typeset — matrices, vectors and Greek letters rendered as real notation — a table of the formulas that matter, defined terms and the mistakes to watch for, worked examples revealed step by step with the reasoning beside each step, practice questions with hints, a quiz, and unsolved problems to do on your own, studied through nineteen modes, from flashcards and drills to timed exams and a continuous read of the whole course as one page.
live - 003
Linear Algebra III
Eigenvalues, Transformations and Orthogonality. The directions a matrix only stretches — characteristic polynomials, eigenvalues and eigenvectors of a 2×2, triangular shortcuts, the trace and determinant checks, diagonalization and powers computed through the eigenbasis. Then geometry read as matrix multiplication: standard matrices, rotations, reflections, scalings and shears, composition, kernel and image. The chapter closes on right angles — orthogonal and orthonormal sets, projection onto a line, the Gram–Schmidt process, QR, orthogonal complements, least squares and the distance from a point to a subspace. Twenty lectures, 240 questions. Every lecture carries a written explanation with the mathematics properly typeset — matrices, vectors and Greek letters rendered as real notation — a table of the formulas that matter, defined terms and the mistakes to watch for, worked examples revealed step by step with the reasoning beside each step, practice questions with hints, a quiz, and unsolved problems to do on your own, studied through nineteen modes, from flashcards and drills to timed exams and a continuous read of the whole course as one page.
live - 004
Linear Algebra IV
Inner Products, Quadratic Forms and Decompositions. Where the subject hands over to its applications. Inner product spaces and their axioms, integrals as products on spaces of functions, induced norms and Cauchy–Schwarz, weighted products and Gram matrices. Quadratic forms written as x-transpose A x, definiteness read off the eigenvalues or the leading principal minors, and the spectral theorem. Then the decompositions that do the work in practice — LU, Cholesky, singular values, the full SVD, the pseudoinverse and functions of a matrix. A closing chapter on complex eigenvalues, Markov chains and steady states, abstract vector spaces, change of basis, defective matrices and the Jordan form, and orthogonal matrices. Twenty-two lectures, 264 questions. Every lecture carries a written explanation with the mathematics properly typeset — matrices, vectors and Greek letters rendered as real notation — a table of the formulas that matter, defined terms and the mistakes to watch for, worked examples revealed step by step with the reasoning beside each step, practice questions with hints, a quiz, and unsolved problems to do on your own, studied through nineteen modes, from flashcards and drills to timed exams and a continuous read of the whole course as one page.
live - 005
Linear Algebra V
Complex Matrices, Deep Structure, Numerics and Applications. The capstone course. Complex vectors with the Hermitian inner product, Hermitian and unitary matrices, the spectral theorem in full strength and the normal matrices it exactly covers. Then the deeper structure of a matrix — Cayley–Hamilton, the minimal polynomial, generalized eigenvectors and Jordan form, the matrix exponential and the systems of differential equations it solves. A numerical chapter on vector and matrix norms, the condition number, the power method and optimal low-rank approximation by the Eckart–Young theorem. And a closing chapter of linear algebra at work: Fibonacci and Binet's formula, principal component analysis, graphs and the Laplacian, PageRank with Perron–Frobenius, and least-squares fitting done stably. Twenty lectures, 260 questions. Every lecture carries a written explanation with the mathematics properly typeset — matrices, vectors and Greek letters rendered as real notation — a table of the formulas that matter, defined terms and the mistakes to watch for, worked examples revealed step by step with the reasoning beside each step, practice questions with hints, a quiz, and unsolved problems to do on your own, studied through nineteen modes, from flashcards and drills to timed exams and a continuous read of the whole course as one page.
live - 006
The whole of linear algebra
Every level at once — all five courses, nineteen chapters, 115 lectures and 1,400 questions read as one book. The whole-course view strings every lecture into a single continuous page from the first vector to PageRank, the formula sheet and glossary pool the entire subject, and mixed exams draw questions from all five courses at once. This is the one to open when you want to read linear algebra from the beginning or revise across course boundaries rather than work through a single level.
live