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- https://math.libretexts.org/Bookshelves/Calculus/Calculus_3e_(Apex)/10%3A_Vectors/10.03%3A_The_Dot_ProductThe previous section introduced vectors and described how to add them together and how to multiply them by scalars. This section introduces a multiplication on vectors called the dot product.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/06%3A_OrthogonalityThis page outlines a chapter on solving matrix equations Ax=b, emphasizing orthogonality for approximate solutions. It begins with definitions in Sections 6.1 and 6.2, discusses orthogonal project...This page outlines a chapter on solving matrix equations Ax=b, emphasizing orthogonality for approximate solutions. It begins with definitions in Sections 6.1 and 6.2, discusses orthogonal projections for finding closest vectors in Section 6.3, and introduces the least-squares method in Section 6.5, highlighting its applications in data modeling, including predicting best-fit lines or ellipses in historical astronomical data.
- https://math.libretexts.org/Courses/Mission_College/MAT_04C_Linear_Algebra_(Kravets)/07%3A_Orthogonality/7.03%3A_Orthogonal_ProjectionLet W be a subspace of Rn and let x be a vector in Rn . In this section, we will learn to compute the closest vector xW to x in W. The vector xW is called the orthogonal projection of x...Let W be a subspace of Rn and let x be a vector in Rn . In this section, we will learn to compute the closest vector xW to x in W. The vector xW is called the orthogonal projection of x onto W . This is exactly what we will use to almost solve matrix equations.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Matrix_Analysis_(Cox)/04%3A_Least_Squares/4.01%3A_Least_SquaresWe learned in the previous chapter that Ax=b need not possess a solution when the number of rows of A exceeds its rank, i.e., r<m. As this situation arises quite often in practice, typically in t...We learned in the previous chapter that Ax=b need not possess a solution when the number of rows of A exceeds its rank, i.e., r<m. As this situation arises quite often in practice, typically in the guise of 'more equations than unknowns,' we establish a rationale for the absurdity Ax=b.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/04%3A_R/4.11%3A_Least_Squares_ApproximationIn this section, we discuss a very important technique derived from orthogonal projections: the least squares approximation.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/06%3A_Orthogonality/6.03%3A_Orthogonal_ProjectionThis page explains the orthogonal decomposition of vectors concerning subspaces in Rn, detailing how to compute orthogonal projections using matrix representations. It includes methods f...This page explains the orthogonal decomposition of vectors concerning subspaces in Rn, detailing how to compute orthogonal projections using matrix representations. It includes methods for deriving projection matrices, with an emphasis on linear transformations and their properties. The text outlines the relationship between a subspace and its orthogonal complement, utilizing examples to illustrate projection calculations and reflections across subspaces.
