Linear Algebra
Linear algebra studies vector spaces and the linear maps between them, providing the computational and conceptual backbone for essentially all of quantitative science and a central chapter of abstract algebra.
Definition
Linear algebra is the study of vector spaces over a field and the linear transformations between them, together with the representation of these transformations by matrices and their classification up to equivalence and similarity.
Scope
This area covers vector spaces, bases and dimension, linear transformations and their matrices, kernels and images, eigenvalues and eigenvectors, diagonalization, inner product spaces, the spectral theorem, and canonical forms such as the Jordan and rational canonical forms. It treats both the concrete matrix theory and the coordinate-free structural viewpoint.
Sub-topics
Core questions
- What is the dimension of a vector space and how do bases relate to one another?
- How is a linear transformation represented by a matrix, and how does that change under a change of basis?
- When can a linear operator be diagonalized, and what canonical form does it admit otherwise?
- How do inner products and orthogonality refine the structure of a vector space?
Key theories
- Rank-nullity theorem
- For a linear map between finite-dimensional spaces, the dimension of the domain equals the rank (dimension of the image) plus the nullity (dimension of the kernel), tying together the solvability of linear systems and dimension counting.
- Spectral theorem
- A self-adjoint (or normal) operator on a finite-dimensional inner product space admits an orthonormal basis of eigenvectors and is therefore diagonalizable by a unitary change of basis.
- Jordan and rational canonical forms
- Every linear operator on a finite-dimensional space over a field is similar to a unique canonical matrix (Jordan form over an algebraically closed field, rational canonical form over any field) determined by invariant factors, classifying operators up to similarity.
Clinical relevance
Linear algebra is the workhorse of applied mathematics: it underlies numerical computation, optimization, statistics and regression, quantum mechanics, computer graphics, machine learning, and signal processing, where high-dimensional data and operators are modeled as vectors and matrices.
History
Linear algebra emerged from the study of systems of linear equations and determinants, was given matrix form by Cayley and Sylvester in the mid-nineteenth century, and was abstracted into the theory of vector spaces by Grassmann, Peano, and others. The eigenvalue and spectral theory matured alongside the development of functional analysis and quantum mechanics.
Key figures
- Arthur Cayley
- James Joseph Sylvester
- Camille Jordan
- Hermann Grassmann
- David Hilbert
Related topics
Seminal works
- hoffman1971
- roman2008
- lang2002
Frequently asked questions
- How is linear algebra related to module theory?
- A vector space is precisely a module over a field. Module theory generalizes linear algebra to coefficients in an arbitrary ring, where phenomena such as the lack of a basis appear; canonical-form theory for operators is a special case of the structure theorem for modules over a principal-ideal domain.
- When can a matrix be diagonalized?
- A square matrix is diagonalizable over a field exactly when its minimal polynomial splits into distinct linear factors over that field, equivalently when there is a basis of eigenvectors. Otherwise the closest standard representative is its Jordan or rational canonical form.