Alphabeta Math

Linear Algebra

44 pages in 5 parts

Linear algebra develops vector spaces, linear maps and matrices over fields, then follows the structural and computational consequences of these objects. Core topics include dimension, bases, rank-nullity, change of basis, Gaussian elimination, determinants, bilinear and quadratic forms, inner products and spectral theory. Further topics include complexification and realification, exterior powers and Hodge duality, LU, Cholesky and QR factorizations, Krylov methods, conjugate gradients, MINRES, GMRES and pseudoinverses. Matrix norms, conditioning and regularized least squares connect the theory to numerical stability, while matrix differentiation and perturbation pages analyze changing operators.

Some topics draw on Abstract Algebra, Real Analysis, Topology and Combinatorics. Linear algebra supplies methods used in Abstract Algebra, Representation Theory of Groups, Coxeter Groups, Algebraic Geometry and Differential Topology. Its inner-product, spectral, matrix and subspace tools support Numerical Linear Algebra and Approximation, Quantum Algorithms, Quantum Computing and Statistical Learning and Classical Models. Other applied connections include Machine Learning Algorithms, Neural Networks and Learning Dynamics, and Inverse Problems.

Pathway

The parts run in order. Everything a page needs from this group has been read by the time you reach it, and the level on each row is how many dependency steps into the group that page sits.

  1. Part 1 · Spaces and linear maps

    3 pages

    A vector space is defined over an arbitrary field, and the first results are the ones that need no counting: subspaces, spans, sums and internal direct sums. Linear independence then buys dimension, with every basis of a space having the same size, and a linear map is measured by its kernel and image through rank-nullity and the quotient space.

  2. Part 2 · Matrices and elimination

    2 pages · after Part 1

    A matrix is what a linear map becomes once bases are chosen at both ends, and change of basis is what happens when they are chosen differently. Row reduction is the algorithm that answers the concrete questions, so elementary matrices, reduced row echelon form and the uniqueness of that form carry the computational weight of everything later.

  3. Part 3 · Determinants

    2 pages · after Part 2

    The determinant is built over a commutative ring, from the permutation sum, so that it applies to matrices whose entries are polynomials as well as to matrices of numbers. It is multiplicative, it detects invertibility, and because similar matrices share it, it descends to an endomorphism and gives cofactor expansion and Cramer's rule.

  4. Part 4 · Forms and inner products

    2 pages · after Part 3

    The dual space collects the linear functionals, and a bilinear or quadratic form is classified over the reals by Sylvester's law of inertia, which says the signature is what survives a change of basis. An inner product adds length and angle, and with it come Gram-Schmidt, orthogonal projection, the adjoint of an operator, and the geometry that makes least squares and spectral arguments possible.

  5. Part 5 · Eigenvalues and canonical forms

    13 pages · after Parts 3 and 4

    An eigenvalue is a characteristic-polynomial root, and the minimal polynomial, generalized eigenspaces, Jordan form, scalar-change comparisons, Schur form, singular values, norms, conditioning, and exterior powers organize similarity, orthogonality, and perturbation. LU, Cholesky, QR, the pseudoinverse, and regularized least squares add factorization and inverse-problem techniques, while matrix differentials give Fréchet derivative rules and sharp sensitivity phenomena at defective or crossing spectra. Krylov, Arnoldi, Ritz, and GMRES then turn polynomial approximation into matrix-free solvers. The added Hermitian page specializes that story: conjugate gradients are the energy-minimizing Galerkin method for positive-definite systems, Lanczos is Arnoldi with a three-term recurrence, MINRES handles Hermitian residual minimization, and preconditioning is analyzed through the transformed operator.