Linear Algebra Textbook
From the axioms of vector spaces to Jordan normal form. A textbook series building definitions, theorems, and proofs in a systematic progression.
Linear Algebra: A Complete Summary of Definitions, Theorems, and Proofs
A single-page survey of undergraduate linear algebra. From the axioms of a vector space to Jordan normal form, this article systematically organizes the definitions, theorems, and proofs that form the core of a first course. Includes a topic dependency diagram.
Vector Spaces: Definitions and First Properties
Starting from the eight axioms of a vector space, we prove the uniqueness of the zero vector and basic properties of scalar multiplication. We then develop the theory of subspaces, sum spaces, and direct sums, building up the foundations of linear algebra one step at a time.
Linear Independence, Bases, and Dimension
Beginning with linear combinations and span, we define linear independence, bases, and dimension. The Steinitz exchange lemma establishes that every basis of a finite-dimensional space has the same number of elements, placing dimension on a firm footing.
Linear Maps: Structure-Preserving Maps Between Vector Spaces
We define linear maps and study their fundamental properties, including the kernel and image. The centerpiece is the rank--nullity theorem, which relates the dimensions of these subspaces. We also discuss the vector space of linear maps and the notion of isomorphism.
Matrices and Representation of Linear Maps
We define the matrix representation of a linear map with respect to chosen bases, and show that composition of maps corresponds to matrix multiplication. The change-of-basis formula, the rank of a matrix, invertible matrices, and their properties are developed systematically.
Systems of Linear Equations and Row Reduction
We formulate systems of linear equations in the language of matrices, develop the theory of elementary row operations and reduced row echelon form, and prove the structure theorem for solutions: the general solution is a particular solution plus the null space.
The Determinant: A Scalar Invariant of Square Matrices
Beginning with permutations and their signs, we define the determinant via the Leibniz formula and establish its fundamental properties: multilinearity, alternation, and normalization. We then develop cofactor expansion, the product formula det(AB) = det(A)det(B), the adjugate matrix, and Cramer's rule.
Eigenvalues and Eigenvectors: Invariant Directions of Linear Maps
We define eigenvalues, eigenvectors, and eigenspaces, then develop the characteristic polynomial and prove that similar matrices share the same spectrum. After distinguishing algebraic and geometric multiplicity, we establish the linear independence of eigenvectors for distinct eigenvalues and prove the Cayley--Hamilton theorem.
Diagonalization: Simplifying Matrices by Choice of Basis
We define what it means for a matrix to be diagonalizable and prove the necessary and sufficient conditions in terms of eigenspaces. A step-by-step diagonalization procedure is presented, followed by applications to computing matrix powers. The chapter concludes with Schur's triangularization theorem.
Inner Product Spaces and Orthonormal Bases
Starting from the axiomatic definition of an inner product, we derive the Cauchy--Schwarz inequality and the triangle inequality. We then develop the Gram--Schmidt orthonormalization process, the theory of orthogonal complements, orthogonal projections, and the best-approximation theorem.
Jordan Normal Form: Beyond Diagonalization
For matrices that resist diagonalization, we develop the Jordan normal form as the next-best canonical structure. After introducing Jordan blocks and generalized eigenspaces, we state the existence and uniqueness theorem, explore the connection with the minimal polynomial, and apply the theory to compute the matrix exponential.