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- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/04%3A_R/4.09%3A_Gram-Schmidt_Process/4.9.E%3A_Exercises_for_Section_4.9This page outlines exercises utilizing the Gram-Schmidt process to derive orthonormal bases from various vector sets in R2, R3, and R4. Key exercises in...This page outlines exercises utilizing the Gram-Schmidt process to derive orthonormal bases from various vector sets in R2, R3, and R4. Key exercises include finding bases for pairs and spans of vectors, addressing restrictions, identifying bases for subspaces, and applying the process to different vector sets. Comprehensive solutions accompany each exercise.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/04%3A_R/4.09%3A_Gram-Schmidt_ProcessThe Gram-Schmidt process is an algorithm to transform a set of vectors into an orthonormal set spanning the same subspace, that is generating the same collection of linear combinations.
- https://math.libretexts.org/Under_Construction/Purgatory/Differential_Equations_and_Linear_Algebra_(Zook)/18%3A_Orthonormal_Bases_and_Complements/18.04%3A_Gram-Schmidt_and_Orthogonal_Complementsv^{\perp} \cdot w^{\perp}&=v^{\perp} \cdot \left(w - \dfrac{u\cdot w}{u\cdot u}\,u - \dfrac{v^{\perp}\cdot w}{v^{\perp}\cdot v^{\perp}}\,v^{\perp} \right)\\ &=v^{\perp}\cdot w - \dfrac{u \cdot w}{u \c...v^{\perp} \cdot w^{\perp}&=v^{\perp} \cdot \left(w - \dfrac{u\cdot w}{u\cdot u}\,u - \dfrac{v^{\perp}\cdot w}{v^{\perp}\cdot v^{\perp}}\,v^{\perp} \right)\\ &=v^{\perp}\cdot w - \dfrac{u \cdot w}{u \cdot u}v^{\perp} \cdot u - \dfrac{v^{\perp} \cdot w}{v^{\perp} \cdot v^{\perp}} v^{\perp} \cdot v^{\perp} \\
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/07%3A_Vector_Spaces/7.12%3A_Inner_Product_SpacesThe dot product was introduced in Rn to provide a natural generalization of the geometrical notions of length and orthogonality. The plan in this section is to define an inner product on...The dot product was introduced in Rn to provide a natural generalization of the geometrical notions of length and orthogonality. The plan in this section is to define an inner product on an arbitrary real vector space V (of which the dot product is an example in Rn ) and use it to introduce these concepts in V.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/06%3A_Orthogonality/6.04%3A_The_Method_of_Least_SquaresThis page covers orthogonal projections in vector spaces, detailing the advantages of orthogonal sets and defining the simpler Projection Formula applicable with orthogonal bases. It includes examples...This page covers orthogonal projections in vector spaces, detailing the advantages of orthogonal sets and defining the simpler Projection Formula applicable with orthogonal bases. It includes examples of projecting vectors onto subspaces, emphasizes the importance of orthogonal bases, and introduces the Gram-Schmidt process for generating orthogonal bases from sets of vectors.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Map%3A_Linear_Algebra_(Waldron_Cherney_and_Denton)/14%3A_Orthonormal_Bases_and_Complements/14.04%3A_Gram-Schmidt_and_Orthogonal_Complementsv^{\perp} \cdot w^{\perp}&=v^{\perp} \cdot \left(w - \dfrac{u\cdot w}{u\cdot u}\,u - \dfrac{v^{\perp}\cdot w}{v^{\perp}\cdot v^{\perp}}\,v^{\perp} \right)\\ &=v^{\perp}\cdot w - \dfrac{u \cdot w}{u \c...v^{\perp} \cdot w^{\perp}&=v^{\perp} \cdot \left(w - \dfrac{u\cdot w}{u\cdot u}\,u - \dfrac{v^{\perp}\cdot w}{v^{\perp}\cdot v^{\perp}}\,v^{\perp} \right)\\ &=v^{\perp}\cdot w - \dfrac{u \cdot w}{u \cdot u}v^{\perp} \cdot u - \dfrac{v^{\perp} \cdot w}{v^{\perp} \cdot v^{\perp}} v^{\perp} \cdot v^{\perp} \\
