🔢Linear Algebra — AristoMind Academy

Master vectors, linear maps, eigenvalues, and the spectral theorem — with geometric intuition and problem-focused learning.

Linear Algebra is the language of modern mathematics, physics, machine learning, and engineering. Every chapter includes worked examples, exam-style problems, and a concise exam summary so you know exactly what to learn before a test.

Whether you're encountering vector spaces for the first time, already deep in a university course, or revisiting the foundations for higher-level work in data science or theoretical physics — this book gives you the geometric intuition, careful derivations, and step-by-step solutions needed to truly master the subject.

📘 What You’ll Learn

📐 Vectors & Geometry — Vectors in Rⁿ, dot products, norms, unit vectors, and geometric applications including lines, planes, and spheres in R² and R³.

🏗️ Vector Spaces — Subspaces, linear combinations, span, linear independence, bases, dimension, inner product spaces, and orthonormal bases.

🔄 Linear Maps & Matrices — Kernel, image, injectivity, surjectivity, rank, nullity, and the matrix of a linear map.

✖️ Matrix Operations — Addition, scalar multiplication, matrix multiplication, composition of linear maps, and the relationship between composition and matrix products.

🔁 Isomorphisms & Inverses — Invertible functions, basis criteria for isomorphisms, finding matrix inverses (including the general 2×2 formula), change of basis, and the transpose.

🔍 Eigenvalues & Eigenvectors — Characteristic polynomials, determinants (2×2 and 3×3), eigenspaces, the trace, and diagonalisation.

📊 Eigenvalues in Depth — Similar matrices, eigenvalues of linear maps in both finite and infinite dimensions, and the contrast between those two settings.

✨ The Spectral Theorem — Orthogonality, the Gram–Schmidt process, QR factorisation, orthogonal projections onto lines and subspaces, and the least squares method.

🌐 Applications — Systems of differential equations, Markov chains and transport planning, principal component analysis, PageRank, vibration analysis in structural engineering, and population dynamics via Leslie matrices.

📝 Practice & Exam Prep — End-of-chapter problems with solutions and concise exam summaries per chapter.

📚 What's Included

✅ 310+ beautifully formatted pages
✅ 160+ fully solved problems
✅ 370+ illustrations and diagrams
✅ Exam-focused summaries for each chapter
✅ Clickable boxed equations for instant reference

💡 Why AristoMind?

This course was created by someone who's been through the same journey — trained at Oxford and now pursuing a PhD in theoretical physics at Queen Mary University of London, a leading research institute in the field. His work focuses on scattering amplitudes and their connection to black hole physics. You can learn more on the About Us page and the FAQ page.

"From Physics Olympiad training to my years at Oxford and now deep into PhD research, one lesson has remained constant: real understanding comes from solving problems, not memorizing facts. That's why this course focuses on clear explanations, step-by-step solutions, and exam-focused summaries — to help you truly grasp and retain the material.

It's designed for students in physics, engineering, or mathematics who want to build clarity and confidence as they learn." — Daniel Siretanu, author

📦 Format & Delivery

• 📥 Instantly available: Access PDF directly after purchase on your favourite device.

🚀 Upgrade your math toolkit today!

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