An orthogonal basis of a vector space is a set of vectors that are mutually perpendicular and span the entire space, allowing any vector in the space to be uniquely represented as a linear combination of these basis vectors. This concept simplifies many mathematical computations, such as projections and transformations, due to the orthogonality property that enables easy calculation of coefficients in the linear combination.

Orthogonal Basis

orthogonal basis

A vector space is a mathematical structure formed by a collection of vectors, which can be added together and multiplied by scalars, adhering to specific axioms such as associativity, commutativity, and distributivity. It provides the foundational framework for linear algebra, enabling the study of linear transformations, eigenvalues, and eigenvectors, which are crucial in various fields including physics, computer science, and engineering.

vector space

mutually perpendicular

span the entire space

A linear combination involves summing multiple vectors, each multiplied by a scalar coefficient, to form a new vector in the same vector space. This concept is fundamental in linear algebra and is used in various applications such as solving linear equations, transformations, and understanding vector spaces and their spans.

linear combination

mathematical computations

projections

Transformations refer to operations that alter the form, appearance, or structure of something, often used in mathematics and science to describe changes in position, shape, or size while preserving certain properties. They are fundamental in understanding symmetry, geometry, and the behavior of functions and systems across various domains.

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