linear-algebra
Installation
SKILL.md
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)