Linear algebra

Linear algebra studies vector spaces, linear transformations, and systems of linear equations. It provides tools for representing and solving problems involving linear relationships.

Vector: A mathematical object with magnitude and direction, or an element of a vector space. Vectors are the objects that linear combinations and transformations act on.
Gaussian elimination: An algorithm that solves linear systems by applying row operations to a matrix. It reduces a system to a form where its solutions can be read or classified.
Least squares: A method for fitting parameters by minimizing the sum of squared residuals. When equations are inconsistent, least squares finds the solution with the smallest residual error.
Nonlinear system: A system of equations or transformations in which at least one relation is nonlinear. Nonlinear relations generally cannot be handled by the same direct methods as linear systems.
Matrix: A rectangular array of numbers or other entries arranged in rows and columns. Matrices encode linear systems and represent linear transformations in chosen bases.
Linear transformation: A function between vector spaces that preserves vector addition and scalar multiplication. Linear transformations are the structure-preserving maps at the heart of the subject.
Principal component analysis: A statistical method that projects data onto orthogonal directions of greatest variance. It uses eigenvectors of a covariance matrix to reduce data dimensions.
Abstract algebra: The study of algebraic structures such as groups, rings, and fields. Linear algebra uses algebraic structures but focuses specifically on vector spaces and linear maps.
Vector space: A set whose elements can be added and scaled while satisfying specified algebraic rules. Vector spaces supply the abstract setting for vectors, bases, and linear maps.
Rank–nullity theorem: A theorem stating that a linear map’s domain dimension equals its rank plus its nullity. It links the independent outputs of a map to the dimensions of its kernel and domain.

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