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)