1. Vectors and Vector Spaces
2. Linear Independence, Span, Basis, and Dimension
3. Matrices
4. Systems of Linear Equations
5. Eigenvalues and Eigenvectors
6. Linear Transformations
7. Inner Product Spaces
8. Singular Value Decomposition
9. Problem Set

University-level linear algebra notes covering vector spaces, matrices, and decompositions.

• Vector Spaces and Subspaces: Definitions, examples, bases, dimension
• Matrices and Systems: Matrix operations, determinants, linear systems
• Eigenvalues and Eigenvectors: Diagonalisation, characteristic polynomials
• Linear Transformations: Kernel, image, rank-nullity theorem

• Basic algebra and arithmetic
• Mathematical proofs and logic
• Familiarity with mathematical notation

Start with vector spaces to build foundational knowledge, then progress to matrices and decompositions. Each section includes worked examples and practice problems.

Use the sidebar to browse topics, or start with the introductory pages linked from the sidebar.

Each section includes:

• Detailed explanations of key concepts
• Worked examples with step-by-step solutions
• Practice problems with answers
• Common pitfalls and how to avoid them
• Connections to other areas of mathematics

1. Master the definitions: Linear algebra requires precise understanding of vector spaces and linear maps
2. Practise proofs: Learn to write clear, rigorous proofs
3. Draw diagrams: Visualise vector spaces, transformations, and decompositions
4. Learn standard examples: Know the properties of common matrices (diagonal, symmetric, orthogonal)
5. Connect to applications: Relate linear algebra to data science, physics, and engineering