Linear Algebra II: Vector Spaces and Linear Transformations

Deepen your mathematical foundation by mastering the structures that power modern engineering, data science, and physics. This comprehensive written course takes you beyond basic matrix arithmetic to explore the elegant world of abstract vector spaces and linear mappings. You will learn to think like a mathematician and solve complex algebraic problems with confidence. Through clear explanations and structured exercises, you will transition from concrete calculations to abstract reasoning. What you'll learn: 1. Understand vector spaces, subspaces, linear independence, spanning sets, and bases. 2. Analyze linear transformations, including their kernel, range, rank, and matrix representations. 3. Calculate eigenvalues, eigenvectors, and perform matrix diagonalization. 4. Explore inner product spaces, orthogonality, and the Gram-Schmidt process. 5. Apply core linear algebra concepts to modern domains like machine learning and data analysis. We begin with essential definitions and foundational axioms before moving on to transformations, coordinates, and spectral theory. This course is designed for university students, self-taught programmers, and exam candidates who have a basic familiarity with introductory matrix operations. Start mastering abstract linear algebra today.