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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.
6.2: Subspaces and Spanning Sets
https://math.libretexts.org/Bookshelves/Linear_Algebra/Linear_Algebra_with_Applications_(Nicholson)/06%3A_Vector_Spaces/6.02%3A_Subspaces_and_Spanning_Sets
Subspaces of a Vector Space018059 If $V$ is a vector space, a nonempty subset $U \subseteq V$ is called a subspace of $V$ if $U$ is itself a vector space using the addition and scalar multipli...Subspaces of a Vector Space018059 If $V$ is a vector space, a nonempty subset $U \subseteq V$ is called a subspace of $V$ if $U$ is itself a vector space using the addition and scalar multiplication of $V$ . Note that the proof of Theorem [thm:018065] shows that if $U$ is a subspace of $V$ , then $U$ and $V$ share the same zero vector, and that the negative of a vector in the space $U$ is the same as its negative in $V$ .
2.7: Basis and Dimension
https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/02%3A_Systems_of_Linear_Equations-_Geometry/2.07%3A_Basis_and_Dimension
This page discusses the concept of a basis for subspaces in linear algebra, emphasizing the requirements of linear independence and spanning. It covers the basis theorem, providing examples of finding...This page discusses the concept of a basis for subspaces in linear algebra, emphasizing the requirements of linear independence and spanning. It covers the basis theorem, providing examples of finding bases in various dimensions, including specific cases like planes defined by equations. The text explains properties of subspaces such as the column space and null space of matrices, illustrating methods for finding bases and verifying their dimensions.
4.6: Subspaces and Bases
https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/04%3A_R/4.06%3A_Subspaces_and_Bases
The goal of this section is to develop an understanding of a subspace of $\mathbb{R}^n$ .
4.9.E: Exercises for Section 4.9
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_4.9
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 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.
7.5: Subspaces
https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/07%3A_Vector_Spaces/7.05%3A_Subspaces
In this section we will examine the concept of subspaces introduced earlier in terms of Rn. Here, we will discuss these concepts in terms of abstract vector spaces.
14.1: Subspaces
https://math.libretexts.org/Under_Construction/Purgatory/Differential_Equations_and_Linear_Algebra_(Zook)/14%3A_Subspaces_and_Spanning_Sets/14.01%3A_Subspaces
A subspace of a vector space V is a subset U under the inherited addition and scalar multiplication operations of V .
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.
6.2: Orthogonal Complements
https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/06%3A_Orthogonality/6.02%3A_Orthogonal_Complements
This page explores orthogonal complements in linear algebra, defining them as vectors orthogonal to a subspace $W$ in $\mathbb{R}^n$ . It details properties, computation methods (such as using RREF...This page explores orthogonal complements in linear algebra, defining them as vectors orthogonal to a subspace $W$ in $\mathbb{R}^n$ . It details properties, computation methods (such as using RREF), and visual representations in $\mathbb{R}^2$ and $\mathbb{R}^3$ . Key concepts include the relationship between a subspace and its double orthogonal complement, the equality of row and column ranks of matrices, and the significance of dimensions in relation to null spaces.
7.12E: Exercises for Section 7.12
https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/07%3A_Vector_Spaces/7.12%3A_Inner_Product_Spaces/7.12E%3A_Exercises_for_Section_7.12
This page contains exercises on inner product spaces, focusing on identifying properties and verifying definitions. Key activities include analyzing axioms for inner products, demonstrating subspaces,...This page contains exercises on inner product spaces, focusing on identifying properties and verifying definitions. Key activities include analyzing axioms for inner products, demonstrating subspaces, computing distances, and checking properties for functions in $D_n$ . It emphasizes symmetry, linearity, and positive-definiteness while exploring inner products in complex numbers and matrices.
11.4: A.4- Subspaces, Dimension, and The Kernel
https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/MAT-204%3A_Differential_Equations_for_Science_(Lebl_and_Trench)/11%3A_Appendix_A-_Linear_Algebra/11.04%3A_A.4-_Subspaces_Dimension_and_The_Kernel
It is the 3-dimensional space \[\text{column space of $L$} = \operatorname{span} \left\{ $\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$ , $\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$ , \begin{bmatrix} 0 \\ 0 ...It is the 3-dimensional space $\text{column space of $L$} = \operatorname{span} \left\{ \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} , \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} , \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \right\} = {\mathbb{R}}^3 . \nonumber$ The row space is the 3-dimensional space \[\text{row space of $L$} = \operatorname{span} \left\{ $\begin{bmatrix} 1 & 2 & 0 & 0 & 3 \end{bmatrix}$ , $\begin{bmatrix} 0 & 0 & 1 & 0 & 4 \end{bmatrix}$ , \begin{bmatrix} 0 & 0 & 0 & 1 & 5 \end{bma…

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