Linear Algebra¶
Teaching material designed and developed by Fabio Furini, associate professor at DIAG, Sapienza University of Rome.
The course lecture notes, online: definitions, worked examples, exercises with full solutions, and a computation lab that shows every step — Gaussian elimination, determinants, inverses, linear systems, LU factorization — with the same notation as the notes.
Start with sums Try the lab Download all the lecture notes (PDF)
Matrices, step by step¶
Watch how a matrix becomes upper triangular with simple row operations: press «Next step» and the zeros appear below the diagonal, one step at a time (or ▶ to let it run by itself).
The parts of the course¶
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1–2 · Sums and products
Summation and product notation, their properties, closed-form sums, the factorial.
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3–4 · Vectors and matrices
Vectors and linear independence, matrices, determinants and rank; elementary operations and Gaussian elimination, inverse matrix, LU factorization, eigenvalues and Sylvester's criterion.
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5 · Norms
The ℓ₁, ℓ₂, ℓ∞ and generalized ℓ₂ norms, the Cauchy–Schwarz inequality and the triangle inequality.
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6 · Linear systems
Existence and uniqueness of solutions, the Rouché–Capelli theorem, Gaussian elimination, homogeneous systems, the LU method, Cramer's rule.
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A · Further topics
Absolute value and modular arithmetic.
How each chapter is organized¶
Each chapter is one of the course notes, with the same numbers and the same colours as the PDF: on the website and on paper you find everything in the same place.
Definition
The precise meaning of a new concept.
Observation, Proposition, Theorem
A result to remember, with its hypotheses. In these notes most results are Observations: they share the theorems' red box and often come with a proof.
Example
A computation worked out in full, one step at a time.
In green the key point to take away: rules, methods, summaries.
Proof — opens with a click
All proofs are there, but folded: first read the statement, then open the proof when you want to study it.
Try it in the lab
In the chapters with computations, the same example can be redone step by step in the lab: change the matrix and watch the steps change.
Don't miss¶
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The computation lab
Eleven tools that show the steps, the way you would write them by hand, with exercises generated for you.
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Exercises with solutions
One sheet per chapter, with the full solution one click away.
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The course
How to use the site, where these topics come back in the Operations Research courses, the full index of the chapters.
By the same author:
- Linear Programming — polyhedra, the simplex algorithm, duality and the dual simplex, with the interactive lab;
- Convex Optimization — convex sets and functions, convex optimization problems, with interactive graphs;
- Algorithms — number representation, pseudo-code, sorting and searching, analysis and recursive algorithms;
- Integer Linear Programming — LP relaxation, valid inequalities, cutting planes, branch-and-bound and branch-and-cut;
- Combinatorial Optimization — knapsack, bin packing, coloring, TSP, scheduling and lot sizing, solved in every way;
- Network Optimization — graphs, spanning trees, shortest paths, network flows, the network simplex and project management;
- MIP Modelling — integer linear programming models, from the formulation to the solution with solvers;
- Operations Research Lab — modelling and solving optimization problems with Python and solvers;
- Mathematical Analysis 1 — real numbers, functions, limits, derivatives and series, with interactive graphs.
Teaching material by Fabio Furini — DIAG, Sapienza University of Rome.
Questo sito è disponibile anche in italiano.