Linear algebra plays a fundamental role in topics as varied as machine learning, quantum mechanics, and computer graphics. A "syntax of space," linear algebra describes the rules for transforming shapes that can be modeled as vectors and matrices. ![[DALLE3_LinearAlgebra.png]] ## Concept Tree ### Basic 1. **Matrices and Vectors** - Definition and notation - Matrix and vector addition - Scalar multiplication 2. **Matrix Multiplication** - Matrix-vector multiplication - Matrix-matrix multiplication 3. **Special Types of Matrices** - Diagonal matrices - Transpose of a matrix - Identity matrices ### Advanced 1. **Determinants** - Definition and properties - Cofactor expansion 2. **Inverse Matrices** - Finding the inverse - Properties of the inverse 3. **Eigenvalues and Eigenvectors** - Definition and properties - Calculation methods 4. **Linear Transformations** - Definition and examples - Matrix representation of linear transformations ### Mastery 1. **Vector Spaces** - Subspaces - Basis and dimension - Row space and column space 2. **Orthogonality** - Orthogonal sets and bases - Gram-Schmidt process 3. **Least Squares and Applications** - Least squares solutions - Applications in regression and data fitting 4. **Applications in Computer Science and AI** - Image transformations - Principal Component Analysis (PCA) - Singular Value Decomposition (SVD)