linear algebra

Linear algebra is the branch of mathematics devoted to the theory oflinear structure. The axiomatic treatment of linear structure isbased on the notions of a linear space (more commonly known as avector space), and a linear mapping. Broadly speaking,there are two fundamental questions considered by linear algebra:

• the solution of a linear equation, and

• diagonalization, a.k.a. the eigenvalue problem.

From the geometric point of view, “linear” is synonymous with“straight”, and consequently linear algebra can be regarded as thebranch of mathematics dealing with lines and planes, as well as withtransformations of space that preserve “straightness”, e.g.rotations and reflections. The two fundamental questions, in geometricterms, deal with

• the intersection of hyperplanes, and

• the principal axes of an ellipsoid.

Linearity is a very basic notion, and consequently linear algebra hasapplications in numerous areas of mathematics, science, andengineering. Diverse disciplines, such as differential equations,differential geometry, the theory of relativity, quantum mechanics,electrical circuits, computer graphics, and information theory benefitfrom the notions and techniques of linear algebra.

Euclidean geometry is related to a specialized branch of linearalgebra that deals with linear measurement. Here the relevant notionsare length and angle. A typical question is the determination oflines perpendicular to a given plane. A somewhat less specializedbranch deals with affine structure, where the key notion is that ofarea and volume. Here determinants play an essential role.

Yet another branch of linear algebra is concerned with computation,algorithms, and numerical approximation. Important examples of suchtechniques include: Gaussian elimination, the method of least squares,LU factorization, QR decomposition, Gram-Schmidt orthogonalization,singular value decomposition, and a number of iterative algorithms forthe calculation of eigenvalues and eigenvectors.

The following subject outline surveyskey topics in linear algebra.

1. Linear structure.

   1. Introduction: systems of linear equations, Gaussianelimination, matrices, matrix operations.

   2. Foundations: fields and vector spaces, subspace, linearindependence, basis, ordered basis, dimension, direct sum decomposition.

   3. Linear mappings: linearity axioms, kernels and images,injectivity, surjectivity, bijections, compositions, inverses,matrix representations, change of bases, conjugation, similarity.

2. Affine structure.

   1. Determinants: characterizing properties, cofactorexpansion, permutations, Cramer’s rule, classical adjoint.

   2. Geometric aspects: Euclidean volume, orientation,equiaffine transformations, determinants as geometric invariantsof linear transformations.

3. Diagonalization and Decomposition.

   1. Basic notions: eigenvector, eigenvalue, eigenspace,characteristic polynomial.

   2. Obstructions: imaginary eigenvalues, nilpotenttransformations, classification of 2-dimensional realtransformations.

   3. Structure theory: invariant subspaces, Cayley-Hamiltontheorem, Jordan canonical form, rational canonical form.

4. Multi-linearity.

   1. Foundations: vector space dual, bilinearity, bilineartranspose, Gram-Schmidt orthogonalization.

   2. Bilinearity: bilinear forms, symmetric bilinear forms,quadratic forms, signature and Sylvester’s theorem, orthogonaltransformations, skew-symmetric bilinear forms, symplectictransformations.

   3. Tensor algebra: tensor product, contraction, invariantsof linear transformations, symmetry operations.

5. Euclidean and Hermitian structure.

   1. Foundations: inner product axioms, the adjoint operation,symmetric transformations, skew-symmetric transformations,self-adjoint transformations, normal transformations.

   2. Spectral theorem: diagonalization of self-adjointtransformations, diagonalization of quadratic forms.

6. Computational and numerical methods.

   1. Linear problems: LU-factorization, QR decomposition, leastsquares, Householder transformations.

   2. Eigenvalue problems: singular value decomposition, Gauss andJacobi-Siedel iterative algorithms.