Projeção ortogonal é um conceito fundamental em álgebra linear 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.
Matematicamente, para um vetor 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.
Essa projeção minimiza a distância Euclidiana between the original vector v and its projection onto the subspace, ensuring that the resulting vector is as close as possible to v enquanto ainda permanece dentro do subespaço.
Projeções ortogonais são amplamente utilizadas em diversos campos, incluindo gráficos computacionais, 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 Análise de Componentes Principais (PCA), where data is projected onto lower-dimensional subspaces to reveal patterns and structures.