La proyección ortogonal es un concepto fundamental en álgebra lineal and geometry, used to project vectors onto a specific subspace. In simpler terms, it refers to the process of dropping a perpendicular (orthogonal) line from a point (or vector) to a line (or plane) in a vector space, resulting in the closest point in that line or plane.
Matemáticamente, para un vector v in a vector space and a subspace defined by an base ortonormal, the orthogonal projection is calculated using the formula:
projW(v) = Σ (v · wi) wi, where wi are the orthonormal basis vectors of the subspace W.
Esta proyección minimiza la Distancia Euclidiana between the original vector v and its projection onto the subspace, ensuring that the resulting vector is as close as possible to v mientras aún se encuentra dentro del subespacio.
Las proyecciones ortogonales se utilizan ampliamente en diversos campos, incluyendo gráficos por computadora, where they help in rendering scenes by projecting 3D points onto 2D planes for display on screens. Additionally, they play a significant role in data science, particularly in dimensionality reduction techniques such as Análisis de componentes principales (PCA), where data is projected onto lower-dimensional subspaces to reveal patterns and structures.