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Linear algebra is a branch of mathematics that studies vector spaces, linear mappings between such spaces, and systems of linear equations. It is a fundamental tool in many areas of mathematics and has applications in numerous scientific and engineering disciplines. Key concepts in linear algebra include vectors, matrices, vector spaces, linear transformations, and eigenvalues/eigenvectors.

Vectors are mathematical objects that have both magnitude and direction, often represented as a list of numbers. Vector spaces are collections of vectors that satisfy certain properties, allowing for operations like addition and scalar multiplication. Matrices are rectangular arrays of numbers that can represent linear transformations or systems of linear equations. A linear transformation is a function between vector spaces that preserves vector addition and scalar multiplication. Eigenvalues and eigenvectors are special scalars and vectors, respectively, associated with a linear transformation, revealing important properties of its behavior.

Systems of linear equations can be represented and solved using matrix methods. Techniques such as Gaussian elimination are used to find solutions, which can be unique, infinite, or nonexistent. The study of determinants, which are scalar values associated with square matrices, provides information about the invertibility of a matrix and the uniqueness of solutions to linear systems. Linear algebra is essential for understanding concepts in calculus, differential equations, physics, computer graphics, machine learning, and signal processing. Its abstract nature allows for powerful generalizations and applications across diverse fields.