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5.1: The Principle of Mathematical Induction
https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Gentle_Introduction_to_the_Art_of_Mathematics_(Fields)/05%3A_Proof_Techniques_II_-_Induction/5.01%3A_The_Principle_of_Mathematical_Induction
The Principle of Mathematical Induction (PMI) may be the least intuitive proof method available to us. Indeed, at first, PMI may feel somewhat like grabbing yourself by the seat of your pants and lift...The Principle of Mathematical Induction (PMI) may be the least intuitive proof method available to us. Indeed, at first, PMI may feel somewhat like grabbing yourself by the seat of your pants and lifting yourself into the air. Despite the indisputable fact that proofs by PMI often feel like magic, we need to convince you of the validity of this proof technique. It is one of the most important tools in your mathematical kit!
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.
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.
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.
7.7E: 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.
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.
5.6E: Exercises for Section 5.6
https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/05%3A_Linear_Transformations/5.06%3A_Isomorphisms/5.6E%3A_Exercises_for_Section_5.6
This page contains exercises on linear transformations and isomorphisms in vector spaces, focusing on defining transformations from $\mathbb{R}^3$ and $\mathbb{R}^2$ . It covers properties of isomo...This page contains exercises on linear transformations and isomorphisms in vector spaces, focusing on defining transformations from $\mathbb{R}^3$ and $\mathbb{R}^2$ . It covers properties of isomorphisms, proving conditions for transformations, exploring matrix representations, and finding inverses. The content also discusses constructing matrices that uphold the structure of transformations and their inverses, particularly regarding spans of vectors in higher dimensions.
1.3: The n-dimensional vector space V(n)
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Applied_Geometric_Algebra_(Tisza)/01%3A_Algebraic_Preliminaries/1.03%3A_The_n-dimensional_vector_space_V(n)
Thus the above defintion is consistent with the interpretation of a vector as a pair of numbers indicating the amounts of two chemical species present in a mixture, or alternatively, as a point in pha...Thus the above defintion is consistent with the interpretation of a vector as a pair of numbers indicating the amounts of two chemical species present in a mixture, or alternatively, as a point in phase space spanned by the coordinates and momenta of a system of mass points. Note the characteristic “turning around” of the indices as we pass from Equation $\ref{EQ1.3.4}$ to Equation $\ref{EQ1.3.8}$ with a simultaneous interchange of the roles of the old and the new frame.
2.8: Bases as Coordinate Systems
https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/02%3A_Systems_of_Linear_Equations-_Geometry/2.8%3A_Bases_as_Coordinate_Systems
This page explains how a basis in a subspace serves as a coordinate system, detailing methods for computing $\mathcal{B}$ -coordinates and converting to standard coordinates. It illustrates finding a...This page explains how a basis in a subspace serves as a coordinate system, detailing methods for computing $\mathcal{B}$ -coordinates and converting to standard coordinates. It illustrates finding a basis through row reduction, using examples to demonstrate the representation of vectors as linear combinations of basis vectors. Visual aids support the explanations, emphasizing the verification of linear independence and span to confirm a basis.
4.4: Spanning Sets in Rⁿ
https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/04%3A_R/4.04%3A_Spanning_Sets_in_R
By generating all linear combinations of a set of vectors one can obtain various subsets of $\mathbb{R}^{n}$ which we call subspaces. For example what set of vectors in $\mathbb{R}^{3}$ generate t...By generating all linear combinations of a set of vectors one can obtain various subsets of $\mathbb{R}^{n}$ which we call subspaces. For example what set of vectors in $\mathbb{R}^{3}$ generate the $XY$ -plane? What is the smallest such set of vectors can you find? The tools of spanning, linear independence and basis are exactly what is needed to answer these and similar questions and are the focus of this section.

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