Orthogonality

Norms: A norm is a mathematical concept that measures the size or length of vectors in a vector space. It is a fundamental tool in linear algebra and is typically used to define distances between vectors.

Dot Product: The dot product is a scalar quantity that can be defined between two vectors. It is fundamental to geometry and can be used to determine angles between vectors and to project vectors onto other vectors.

Inner Products: An inner product is a generalization of the dot product to complex-valued vectors. It is a positive definite bilinear form and can be used to define distances and angles between vectors.

Orthogonal Vectors: Two vectors are said to be orthogonal if the angle between them is 90 degrees. Orthogonal vectors are fundamental to linear algebra and can be used to define bases, solve systems of linear equations, and project vectors onto subspaces.

Orthonormal Vectors: Orthonormal vectors are a set of vectors that are both orthogonal and normalized. These vectors are particularly useful in linear algebra as they make computations easier and can be used to define orthonormal bases.

Gram-Schmidt Process: The Gram-Schmidt process is a method for creating an orthonormal basis from a set of linearly independent vectors. The process involves a series of orthogonalization and normalization steps.

Orthogonal Matrices: An orthogonal matrix is a square matrix whose columns are orthonormal vectors. These matrices are particularly useful in linear algebra as they preserve distances and angles between vectors.

Orthogonal Projections: An orthogonal projection is a linear transformation that projects a vector onto a subspace in a way that preserves the vector's direction and length. These projections are useful in linear algebra as they can be used to solve systems of linear equations, compute eigenvalues and eigenvectors, and perform other linear transformations.

Singular Value Decomposition: The Singular Value Decomposition (SVD) is a fundamental tool in linear algebra that decomposes a matrix into a product of three matrices. The SVD can be used for data compression, image processing, and other applications.

Quadratic Forms: A quadratic form is a real-valued function that is quadratic in its arguments. The function can be represented as a matrix and can be used to define ellipsoids, hyperboloids, and other geometric shapes.

Orthogonal Vectors: Two vectors are said to be orthogonal if their dot product is zero. In other words, they are perpendicular to each other.

Orthogonal Basis: A set of vectors is called an orthogonal basis if all vectors in the set are orthogonal to each other.

Orthonormal Vectors: Two vectors are said to be orthonormal if they are both orthogonal and have a magnitude of 1.

Orthonormal Basis: A set of vectors is called an orthonormal basis if all vectors in the set are orthonormal to each other.

Orthogonal Complement: For a subspace, the orthogonal complement is the set of all vectors that are orthogonal to every vector in the subspace.

Orthogonal Matrix: A square matrix is called orthogonal if its transpose is equal to its inverse.

Orthogonal Projection: The orthogonal projection of a vector onto a subspace is the closest vector in the subspace to the original vector.

Orthogonal Transformation: A linear transformation is called orthogonal if it preserves the dot product between any two vectors.

Orthogonal Diagonalization: A square matrix is said to be orthogonally diagonalizable if it can be diagonalized by an orthogonal matrix.

Orthogonal Operators: In functional analysis, an operator on a Hilbert space is called orthogonal if it preserves the inner product between any two vectors.