Matrix
A rectangular array of numbers arranged in rows and columns. The fundamental object of study in linear algebra with applications in solving systems of equations, transformations, and data representation.
Full article →Linear Algebra
The branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces. Essential for modern physics, engineering, computer science, and data science.
Topics in Linear Algebra
Vector Space
A collection of vectors that can be added together and multiplied by scalars. The abstract framework underlying all of linear algebra, generalizing the notion of Euclidean space.
Full article →Eigenvalue
A scalar associated with a linear transformation that describes how the transformation scales vectors in a particular direction. Critical for understanding matrix behavior and stability analysis.
Full article →Eigenvector
A non-zero vector that changes at most by a scalar factor when a linear transformation is applied. Together with eigenvalues, they reveal the fundamental structure of linear transformations.
Full article →Determinant
A scalar value computed from a square matrix that encodes information about the matrix's linear transformation. Determines invertibility, orientation, and scaling factor of the transformation.
Full article →Linear Transformation
A mapping between vector spaces that preserves vector addition and scalar multiplication. Represented by matrices, they form the geometric foundation of linear algebra.
Full article →Vector
A mathematical object with both magnitude and direction. The basic element of vector spaces, representing quantities like force, velocity, and displacement in physics.
Full article →Linear Systems
A collection of linear equations involving the same set of variables. Solved using matrices, Gaussian elimination, and Cramer's rule with applications throughout science and engineering.
Full article →Subspace
A subset of a vector space that is itself a vector space under the inherited operations. Includes concepts like span, basis, dimension, and orthogonal complements.
Full article →Basis
A set of linearly independent vectors that span a vector space. Provides a coordinate system for the space, with the number of basis vectors defining the dimension.
Full article →Dimension
The number of vectors in any basis for a vector space. A fundamental invariant characterizing the size and complexity of the space.
Full article →Rank
The dimension of the column space or row space of a matrix. Determines the number of linearly independent rows or columns and the dimension of the image of the linear transformation.
Full article →Null Space
The set of all vectors that a matrix maps to the zero vector. Also called the kernel, it represents the solutions to the homogeneous system Ax = 0.
Full article →Orthogonality
The property of vectors being perpendicular, with dot product equal to zero. Fundamental for projections, least squares, and the Gram-Schmidt process.
Full article →Inner Product
A generalization of the dot product that allows the definition of lengths and angles in abstract vector spaces. Essential for Hilbert spaces and quantum mechanics.
Full article →Gram-Schmidt Process
An algorithm for orthonormalizing a set of vectors in an inner product space. Produces an orthogonal basis from any linearly independent set of vectors.
Full article →Least Squares
A method for finding the best approximate solution to an overdetermined system. Minimizes the sum of squared residuals and is fundamental to regression analysis.
Full article →Singular Value Decomposition
A factorization of a matrix into three matrices revealing its fundamental structure. Widely used in data compression, image processing, and recommendation systems.
Full article →LU Decomposition
A factorization of a matrix into a lower triangular matrix and an upper triangular matrix. Efficient for solving multiple linear systems with the same coefficient matrix.
Full article →QR Decomposition
A factorization of a matrix into an orthogonal matrix Q and an upper triangular matrix R. Used for solving least squares problems and eigenvalue computations.
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