Linear Algebra

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. This topic is fundamental to understanding advanced mathematics and has wide applications in science and engineering.

Key concepts: definitions, theorems, proofs, and worked examples related to Matrix.

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. This topic is fundamental to understanding advanced mathematics and has wide applications in science and engineering.

Key concepts: definitions, theorems, proofs, and worked examples related to Vector Space.

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. This topic is fundamental to understanding advanced mathematics and has wide applications in science and engineering.

Key concepts: definitions, theorems, proofs, and worked examples related to Eigenvalue.

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. This topic is fundamental to understanding advanced mathematics and has wide applications in science and engineering.

Key concepts: definitions, theorems, proofs, and worked examples related to Eigenvector.

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. This topic is fundamental to understanding advanced mathematics and has wide applications in science and engineering.

Key concepts: definitions, theorems, proofs, and worked examples related to Determinant.

A mapping between vector spaces that preserves vector addition and scalar multiplication. Represented by matrices, they form the geometric foundation of linear algebra. This topic is fundamental to understanding advanced mathematics and has wide applications in science and engineering.

Key concepts: definitions, theorems, proofs, and worked examples related to Linear Transformation.

A mathematical object with both magnitude and direction. The basic element of vector spaces, representing quantities like force, velocity, and displacement in physics. This topic is fundamental to understanding advanced mathematics and has wide applications in science and engineering.

Key concepts: definitions, theorems, proofs, and worked examples related to Vector.

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. This topic is fundamental to understanding advanced mathematics and has wide applications in science and engineering.

Key concepts: definitions, theorems, proofs, and worked examples related to Linear Systems.

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. This topic is fundamental to understanding advanced mathematics and has wide applications in science and engineering.

Key concepts: definitions, theorems, proofs, and worked examples related to Subspace.

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. This topic is fundamental to understanding advanced mathematics and has wide applications in science and engineering.

Key concepts: definitions, theorems, proofs, and worked examples related to Basis.

The number of vectors in any basis for a vector space. A fundamental invariant characterizing the size and complexity of the space. This topic is fundamental to understanding advanced mathematics and has wide applications in science and engineering.

Key concepts: definitions, theorems, proofs, and worked examples related to Dimension.

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. This topic is fundamental to understanding advanced mathematics and has wide applications in science and engineering.

Key concepts: definitions, theorems, proofs, and worked examples related to Rank.

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. This topic is fundamental to understanding advanced mathematics and has wide applications in science and engineering.

Key concepts: definitions, theorems, proofs, and worked examples related to Null Space.

The property of vectors being perpendicular, with dot product equal to zero. Fundamental for projections, least squares, and the Gram-Schmidt process. This topic is fundamental to understanding advanced mathematics and has wide applications in science and engineering.

Key concepts: definitions, theorems, proofs, and worked examples related to Orthogonality.

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. This topic is fundamental to understanding advanced mathematics and has wide applications in science and engineering.

Key concepts: definitions, theorems, proofs, and worked examples related to Inner Product.

An algorithm for orthonormalizing a set of vectors in an inner product space. Produces an orthogonal basis from any linearly independent set of vectors. This topic is fundamental to understanding advanced mathematics and has wide applications in science and engineering.

Key concepts: definitions, theorems, proofs, and worked examples related to Gram-Schmidt Process.

A method for finding the best approximate solution to an overdetermined system. Minimizes the sum of squared residuals and is fundamental to regression analysis. This topic is fundamental to understanding advanced mathematics and has wide applications in science and engineering.

Key concepts: definitions, theorems, proofs, and worked examples related to Least Squares.

A factorization of a matrix into three matrices revealing its fundamental structure. Widely used in data compression, image processing, and recommendation systems. This topic is fundamental to understanding advanced mathematics and has wide applications in science and engineering.

Key concepts: definitions, theorems, proofs, and worked examples related to Singular Value 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. This topic is fundamental to understanding advanced mathematics and has wide applications in science and engineering.

Key concepts: definitions, theorems, proofs, and worked examples related to LU 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. This topic is fundamental to understanding advanced mathematics and has wide applications in science and engineering.

Key concepts: definitions, theorems, proofs, and worked examples related to QR Decomposition.