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- https://math.libretexts.org/Courses/Coastline_College/Math_C285%3A_Linear_Algebra_and_Diffrential_Equations_(Tran)/04%3A_R/4.07%3A_The_Dot_ProductThere are two ways of multiplying vectors which are of great importance in applications. The first of these is called the dot product. When we take the dot product of vectors, the result is a scalar. ...There are two ways of multiplying vectors which are of great importance in applications. The first of these is called the dot product. When we take the dot product of vectors, the result is a scalar. For this reason, the dot product is also called the scalar product and sometimes the inner product.
- https://math.libretexts.org/Courses/Lorain_County_Community_College/Book%3A_Precalculus_(Stitz-Zeager)_-_Jen_Test_Copy/11%3A_Applications_of_Trigonometry/11.09%3A_The_Dot_Product_and_ProjectionPreviously, we learned how add and subtract vectors and how to multiply vectors by scalars. In this section, we define a product of vectors.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Map%3A_Linear_Algebra_(Waldron_Cherney_and_Denton)/14%3A_Orthonormal_Bases_and_Complements/14.03%3A_Relating_Orthonormal_Basesw_{1} &=& (w_{1}\cdot u_{1}) u_{1} + \cdots + (w_{1}\cdot u_{n})u_{n}\\ w_{n} &=& (w_{n}\cdot u_{1}) u_{1} + \cdots + (w_{n}\cdot u_{n})u_{n}\\ \sum_{i}(u_{j}^{T} w_{i}) (w_{i}^{T} u_{k})\\ &=& u_{j}^...w_{1} &=& (w_{1}\cdot u_{1}) u_{1} + \cdots + (w_{1}\cdot u_{n})u_{n}\\ w_{n} &=& (w_{n}\cdot u_{1}) u_{1} + \cdots + (w_{n}\cdot u_{n})u_{n}\\ \sum_{i}(u_{j}^{T} w_{i}) (w_{i}^{T} u_{k})\\ &=& u_{j}^{T} \left[\sum_{i} (w_{i} w_{i}^{T}) \right] u_{k} \\ \left(\sum_{i} w_{i} w_{i}^{T}\right)v &=& \left(\sum_{i} w_{i} w_{i}^{T}\right)\left(\sum_{j} c^{j}w_{j}\right) \\ &=& \begin{pmatrix}u_{1}^{T}\\u_{2}^{T}\\u_{3}^{T}\end{pmatrix} \begin{pmatrix}u_{1} & u_{2}& u_{3}\end{pmatrix} \\
- https://math.libretexts.org/Bookshelves/Geometry/Geometry_with_an_Introduction_to_Cosmic_Topology_(Hitchman)/03%3A_Transformations/3.02%3A_InversionInversion offers a way to reflect points across a circle. This transformation plays a central role in visualizing the transformations of non-Euclidean geometry, and this section is the foundation of m...Inversion offers a way to reflect points across a circle. This transformation plays a central role in visualizing the transformations of non-Euclidean geometry, and this section is the foundation of much of what follows.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/06%3A_Orthogonality/6.01%3A_Dot_Products_and_OrthogonalityThis page covers the concepts of dot product, vector length, distance, and orthogonality within vector spaces. It defines the dot product mathematically in \(\mathbb{R}^n\) and explains properties lik...This page covers the concepts of dot product, vector length, distance, and orthogonality within vector spaces. It defines the dot product mathematically in \(\mathbb{R}^n\) and explains properties like commutativity and distributivity. Length is derived from the dot product, and the distance between points is defined as the length of the connecting vector. Unit vectors are introduced, and orthogonality is defined as having a dot product of zero.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/03%3A_Determinants/3.02%3A_Properties_of_Determinants/3.2E%3A_Exercises_for_Section_3.2This page includes exercises on matrix operations, specifically focusing on determinants. It explains how row and column operations affect determinants, discusses properties linked to nilpotent and or...This page includes exercises on matrix operations, specifically focusing on determinants. It explains how row and column operations affect determinants, discusses properties linked to nilpotent and orthogonal matrices, and provides proofs regarding matrix similarities that maintain determinant values.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/06%3A_Spectral_Theory/6.07%3A_Orthogonal_DiagonalizationIn this section we look at matrices that have an orthonormal set of eigenvectors.
- 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 \( \mathbb{R}^2 \), \( \mathbb{R}^3 \), and \( \mathbb{R}^4 \). Key exercises in...This page outlines exercises utilizing the Gram-Schmidt process to derive orthonormal bases from various vector sets in \( \mathbb{R}^2 \), \( \mathbb{R}^3 \), and \( \mathbb{R}^4 \). 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/4.08%3A_Orthogonal_Vectors_and_Matrices/4.8.E%3A_Exercise_for_Section_4.8This page outlines exercises on determining orthogonality and orthonormality of vectors, classifying matrices (symmetric, skew symmetric, orthogonal), and the properties of orthogonal matrices, such a...This page outlines exercises on determining orthogonality and orthonormality of vectors, classifying matrices (symmetric, skew symmetric, orthogonal), and the properties of orthogonal matrices, such as preserving vector lengths.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Linear_Algebra_with_Applications_(Nicholson)/10%3A_Inner_Product_Spaces/10.02%3A_Orthogonal_Sets_of_VectorsThe idea that two lines can be perpendicular is fundamental in geometry, and this section is devoted to introducing this notion into a general inner product space V.
- https://math.libretexts.org/Courses/Lorain_County_Community_College/Book%3A_Precalculus_Jeffy_Edits_3.75/11%3A_Applications_of_Trigonometry/11.09%3A_The_Dot_Product_and_ProjectionPreviously, we learned how add and subtract vectors and how to multiply vectors by scalars. In this section, we define a product of vectors.
