Linear Algebra

These notes follow the core path of Anton and Kaul's Elementary Linear Algebra: start with systems of equations, turn their row operations into matrix algebra, reinterpret matrices as transformations, then move from concrete vectors in $\mathbb{R}^n$ to abstract vector spaces and inner product geometry. The later pages focus on structure: eigenvalues, diagonalization, orthogonality, least squares, numerical methods, and singular value decomposition.

The emphasis is computational and conceptual. Row reduction is treated both as a hand algorithm and as the engine behind rank, independence, bases, inverses, determinants, and solvability. Matrix factorizations such as QR, LU, and SVD are included because they explain how the same ideas are used in numerical work and data applications.

Pages

1. Linear Algebra
2. Systems of Linear Equations
3. Gaussian Elimination
4. Matrices and Matrix Algebra
5. Matrix Inverses and Elementary Matrices
6. Determinants
7. Vectors in Rn
8. Orthogonality in Rn
9. General Vector Spaces
10. Bases, Dimension, and Rank
11. Eigenvalues and Eigenvectors
12. Diagonalization and Similarity
13. Inner Product Spaces
14. Orthogonality, Gram-Schmidt, and QR
15. Least Squares
16. Quadratic Forms and Spectral Theorems
17. Linear Transformations
18. Numerical Linear Algebra
19. Singular Value Decomposition
20. Applications and Modeling

Use the pages in order for a first pass. On review, treat the connections sections as a map: most major theorems reappear under different names in systems, matrix algebra, transformations, and geometry.