linear-algebra

## What I do - Perform matrix operations and factorizations - Solve systems of linear equations - Find eigenvalues and eigenvectors - Compute determinants and inverses - Apply singular value decomposition - Work with vector spaces and linear transformations ## When to use me When solving linear systems, analyzing transformations, or performing dimensionality reduction. ## Key Concepts - **Matrix Multiplication**: (AB)_{ij} = Σ_k A_{ik}B_{kj}, generally non-commutative - **Determinant**: det(A) nonzero implies invertibility; det(AB) = det(A)det(B) - **Eigenvalue Equation**: Av = λv, characteristic polynomial det(A - λI) = 0 - **LU Decomposition**: A = LU for solving Ax = b efficiently - **SVD**: A = UΣV^T, reveals rank, pseudoinverse, and low-rank approximations - **Rank**: Dimension of column/row space; rank(A) = n - nullity(A)