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4.11: Orthogonality
https://math.libretexts.org/Courses/Coastline_College/Math_C285%3A_Linear_Algebra_and_Diffrential_Equations_(Tran)/04%3A_R/4.11%3A_Orthogonality
In this section, we examine what it means for vectors (and sets of vectors) to be orthogonal and orthonormal. First, it is necessary to review some important concepts. You may recall the definitions f...In this section, we examine what it means for vectors (and sets of vectors) to be orthogonal and orthonormal. First, it is necessary to review some important concepts. You may recall the definitions for the span of a set of vectors and a linear independent set of vectors.
6.1: Dot Products and Orthogonality
https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/06%3A_Orthogonality/6.01%3A_Dot_Products_and_Orthogonality
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 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.
6: Orthogonality
https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/06%3A_Orthogonality
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 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.
5.3: Orthogonality
https://math.libretexts.org/Bookshelves/Linear_Algebra/Linear_Algebra_with_Applications_(Nicholson)/05%3A_Vector_Space_R/5.03%3A_Orthogonality
\[\begin{aligned} \|\mathbf{x} + \mathbf{y}\|^2 &= (\mathbf{x} + \mathbf{y})\bullet (\mathbf{x} + \mathbf{y}) = \mathbf{x}\bullet \mathbf{x} + \mathbf{x}\bullet \mathbf{y} + \mathbf{y}\bullet \mathbf{... $\begin{aligned} \|\mathbf{x} + \mathbf{y}\|^2 &= (\mathbf{x} + \mathbf{y})\bullet (\mathbf{x} + \mathbf{y}) = \mathbf{x}\bullet \mathbf{x} + \mathbf{x}\bullet \mathbf{y} + \mathbf{y}\bullet \mathbf{x} + \mathbf{y}\bullet \mathbf{y} \\ &= \|\mathbf{x}\|^2 + 2(\mathbf{x}\bullet \mathbf{y}) + \|\mathbf{y}\|^2\end{aligned} \nonumber$
4.11: Orthogonality
https://math.libretexts.org/Bookshelves/Linear_Algebra/A_First_Course_in_Linear_Algebra_(Kuttler)/04%3A_R/4.11%3A_Orthogonality
In this section, we examine what it means for vectors (and sets of vectors) to be orthogonal and orthonormal. First, it is necessary to review some important concepts. You may recall the definitions f...In this section, we examine what it means for vectors (and sets of vectors) to be orthogonal and orthonormal. First, it is necessary to review some important concepts. You may recall the definitions for the span of a set of vectors and a linear independent set of vectors.
8.1: Eigenvalue Problems for y'' + λy = 0
https://math.libretexts.org/Courses/Mission_College/Math_4B%3A_Differential_Equations_(Kravets)/08%3A_Boundary_Value_Problems_and_Fourier_Expansions/8.01%3A_Eigenvalue_Problems_for_y''__y__0
This section deals with five boundary value problems for the differential equation y'' + λy = 0. They are related to problems in partial differential equations that will be discussed in Chapter 9. We ...This section deals with five boundary value problems for the differential equation y'' + λy = 0. They are related to problems in partial differential equations that will be discussed in Chapter 9. We define what is meant by eigenvalues and eigenfunctions of the boundary value problems, and show that the eigenfunctions have a property called orthogonality.
8.1: Eigenvalue Problems for y'' + λy = 0
https://math.libretexts.org/Courses/Mission_College/Math_4B%3A_Differential_Equations_(Reed)/08%3A_Boundary_Value_Problems_and_Fourier_Expansions/8.01%3A_Eigenvalue_Problems_for_y''__y__0
This section deals with five boundary value problems for the differential equation y'' + λy = 0. They are related to problems in partial differential equations that will be discussed in Chapter 9. We ...This section deals with five boundary value problems for the differential equation y'' + λy = 0. They are related to problems in partial differential equations that will be discussed in Chapter 9. We define what is meant by eigenvalues and eigenfunctions of the boundary value problems, and show that the eigenfunctions have a property called orthogonality.
4.4: Orthogonality and Normalization
https://math.libretexts.org/Bookshelves/Differential_Equations/Partial_Differential_Equations_(Walet)/04%3A_Fourier_Series/4.04%3A_Orthogonality_and_Normalization
\[ \begin{align} \int_{-L}^L \cos\bigg(\frac{m\pi x}{L}\bigg) \cdot \cos\bigg(\frac{n\pi x}{L}\bigg) dx & = \frac{1}{2}\int_{-L}^L \cos\bigg(\frac{(m+n)\pi x}{L}\bigg) + \cos\bigg(\frac{(m-n)\pi x}{L}... $\begin{align} \int_{-L}^L \cos\bigg(\frac{m\pi x}{L}\bigg) \cdot \cos\bigg(\frac{n\pi x}{L}\bigg) dx & = \frac{1}{2}\int_{-L}^L \cos\bigg(\frac{(m+n)\pi x}{L}\bigg) + \cos\bigg(\frac{(m-n)\pi x}{L}\bigg) dx\\[6px] & = \bigg\{ \begin{array}{lr} 0 & \mbox{if } n \neq m \\ L & \mbox{if } n=m \end{array}\end{align}, \nonumber$
7.12: Inner Product Spaces
https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/07%3A_Vector_Spaces/7.12%3A_Inner_Product_Spaces
The dot product was introduced in $\mathbb{R}^n$ 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 $\mathbb{R}^n$ 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 $\mathbb{R}^n$ ) and use it to introduce these concepts in $V$ .
4.11: Orthogonality
https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/A_First_Journey_Through_Linear_Algebra/04%3A_R/4.11%3A_Orthogonality
In this section, we examine what it means for vectors (and sets of vectors) to be orthogonal and orthonormal. First, it is necessary to review some important concepts. You may recall the definitions f...In this section, we examine what it means for vectors (and sets of vectors) to be orthogonal and orthonormal. First, it is necessary to review some important concepts. You may recall the definitions for the span of a set of vectors and a linear independent set of vectors.
7: Vector Spaces
https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/07%3A_Vector_Spaces
This page defines vector spaces and describes their properties, including operations, spanning sets, linear independence, and subspaces. It covers bases, subspace operations, linear transformations, a...This page defines vector spaces and describes their properties, including operations, spanning sets, linear independence, and subspaces. It covers bases, subspace operations, linear transformations, and the concepts of image and kernel. The text also discusses the matrix representation of linear transformations and introduces inner product spaces, which apply geometric concepts of length and orthogonality to general vector spaces.

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