Linear algebra

A linear equation relates variables without multiplying variables by one another or raising them to powers greater than one. Several such equations can be treated together as a system. In matrix notation, the system is written Ax = b: A is the matrix of coefficients, x is the vector of unknowns and b records the right-hand sides. This notation makes it possible to study many equations at once. Methods such as row reduction reveal whether a system has no solution, one solution or many solutions, and help compute its solutions.

A vector is often drawn as an arrow or represented by a list of coordinates, but the idea is more general. A vector space is a collection of objects that can be added and multiplied by scalars while obeying consistent rules. Depending on the problem, its vectors might represent directions, numerical data, polynomials or solutions to equations. A linear transformation is a function between vector spaces that preserves addition and scalar multiplication: T(u + v) = T(u) + T(v), and T(c v) = c T(v). In finite-dimensional spaces, once bases are chosen, such a transformation can be represented by a matrix. Matrix multiplication then describes the composition of transformations, with order generally mattering.

A basis is a linearly independent set of vectors that spans a vector space; for a finite-dimensional space, the number of vectors in a basis is its dimension. These ideas describe the structure of a space without requiring every vector to be listed. Linear algebra also studies eigenvectors: if a nonzero vector v is transformed into a scalar multiple of itself, so that T(v) = λv, then v is an eigenvector and λ is its eigenvalue. Eigenvalues and eigenvectors help characterize transformations and are important in many mathematical and scientific problems.

Not every system of equations has an exact solution. When equations are inconsistent—for example, when measurements do not agree perfectly—the method of least squares finds a solution that minimizes the sum of squared residuals. This supports best-fit calculations and data analysis. More broadly, linear algebra connects geometric ideas with algebraic computation and appears in mathematical physics, coding theory, engineering, economics and the social sciences. It is also a building block for areas such as numerical computation, statistics and computer graphics, although real-world models may require methods beyond linear ones.