Linear algebra is a branch of mathematics that studies vectors, vector spaces (or linear spaces), linear transformations, and finite-dimensional linear equations. Vector spaces are an important topic in modern mathematics; linear algebra is widely used in abstract algebra and functional analysis; linear algebra can be specifically represented through analytic geometry. The theory of linear algebra has been generalized into operator theory. Since nonlinear models in scientific research can usually be approximated as linear models, linear algebra is widely used in natural sciences, engineering, computer science, and social sciences.

In mathematics, a matrix refers to a two-dimensional data table arranged vertically and horizontally. It originated from the square matrix composed of coefficients and constants of a set of equations. This concept was first proposed by the British mathematician Kelly in the 19th century.

Matrices are common tools in advanced algebra and are also common in applied mathematics disciplines such as statistical analysis. In physics, matrices are used in circuits, mechanics, optics, and quantum physics; in computer science, matrices are also needed in 3D animation production. Matrix operations are an important issue in the field of numerical analysis. Decomposing a matrix into a combination of simple matrices can simplify matrix operations in theory and practical applications. For some widely used and special matrices, such as sparse matrices and quasi-diagonal matrices, there are specific fast operation algorithms. For the development and application of matrix-related theories, please refer to matrix theory. Infinite-dimensional matrices also appear in fields such as astrophysics and quantum mechanics, which are a generalization of matrices.