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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 $\mathbb{R}^n$ .
4.8.E: Exercise for Section 4.8
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_4.8
This 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.
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.
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.
7.7E: Exercises for Section 7.7
https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/07%3A_Vector_Spaces/7.07%3A_Sums_and_Intersections/7.7E%3A_Exercises_for_Section_7.7
This page discusses exercises on vector spaces and subspaces in $\mathbb{R}^n$ , focusing on geometric interpretations, dimensions of sums and intersections of subspaces $U$ and $W$ , and specific...This page discusses exercises on vector spaces and subspaces in $\mathbb{R}^n$ , focusing on geometric interpretations, dimensions of sums and intersections of subspaces $U$ and $W$ , and specific examples of bases. It clarifies that $U + W$ may span the entire space or have distinct dimensions, while intersections can vary in dimension.
5.5E: Exercises for Section 5.5
https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/05%3A_Linear_Transformations/5.05%3A_One-to-One_and_Onto_Transformations/5.5E%3A_Exercises_for_Section_5.5
This page features exercises on linear transformations and their matrix representations, focusing on properties such as injectivity and surjectivity. It includes tasks to analyze various matrix sizes ...This page features exercises on linear transformations and their matrix representations, focusing on properties such as injectivity and surjectivity. It includes tasks to analyze various matrix sizes and examines the relationship between rank, linear independence, and the properties of transformations. Overall, the content emphasizes essential linear algebra concepts related to the effects of transformations on vector spaces.
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.
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.
5.1: Linear Span
https://math.libretexts.org/Bookshelves/Linear_Algebra/Book%3A_Linear_Algebra_(Schilling_Nachtergaele_and_Lankham)/05%3A_Span_and_Bases/5.01%3A_Linear_Span
The linear span (or just span) of a set of vectors in a vector space is the intersection of all subspaces containing that set. The linear span of a set of vectors is therefore a vector space.
9.2: Spanning Sets
https://math.libretexts.org/Bookshelves/Linear_Algebra/A_First_Course_in_Linear_Algebra_(Kuttler)/09%3A_Vector_Spaces/9.02%3A_Spanning_Sets
In this section we will examine the concept of spanning introduced earlier in terms of Rn . Here, we will discuss these concepts in terms of abstract vector spaces.
2.3: Matrix Equations
https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/02%3A_Systems_of_Linear_Equations-_Geometry/2.03%3A_Matrix_Equations
This page explores the matrix equation $Ax = b$ , defining key concepts like consistency conditions, the relationship between matrix and vector forms, and the significance of spans. It explains that ...This page explores the matrix equation $Ax = b$ , defining key concepts like consistency conditions, the relationship between matrix and vector forms, and the significance of spans. It explains that for $Ax = b$ to have solutions, the vector $b$ must lie within the span of $A$ 's columns. Systems have solutions for all $b$ if $A$ has a pivot in every row.

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