Alphabeta Math

Linear Algebra

44 pages in 5 parts

Linear algebra here is done over an arbitrary field throughout. Vector spaces, subspaces, spans and internal direct sums come first, then linear independence buys dimension, with every basis having the same size, and a linear map is measured by its kernel and image through rank-nullity and the quotient space. A matrix is what a linear map becomes once bases are chosen at both ends, change of basis is what happens when they are chosen differently, and Gaussian elimination, elementary matrices and reduced row echelon form carry the computation. The determinant is built from the permutation sum over a commutative ring rather than a field, so that it applies to matrices of polynomials, which the characteristic polynomial needs; it is multiplicative, it detects invertibility, and it descends to an operator through cofactor expansion and Cramer's rule. Dual spaces, bilinear and quadratic forms and Sylvester's law of inertia follow, then inner products with Gram-Schmidt, orthogonal projection and the adjoint. The collection closes on eigenvalues and the characteristic polynomial, the minimal polynomial and diagonalisability, generalised eigenspaces, triangularisation and Jordan canonical form.

The multivariable calculus of Real Analysis is the nearest consumer, since the total derivative is a linear map and the change of variables factor is a determinant. Measure theory reserves the determinant and elimination pages for the linear change of variables, complex analysis the inner product and spectral material, differential geometry rank-nullity and eigenvalues, and the algebraic blocks of combinatorics the inner product geometry, and representation theory begins from the eigenvalue theory.

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.