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- 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_7.7This 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.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/01%3A_Systems_of_Equations/1.05%3A_Applications_to_Physics/1.5E%3A_Exercises_for_Section_1.6This page outlines exercises that analyze electrical circuits using systems of equations from Kirchhoff's laws, involving diagrams and components like resistors and voltage sources. The objective is t...This page outlines exercises that analyze electrical circuits using systems of equations from Kirchhoff's laws, involving diagrams and components like resistors and voltage sources. The objective is to compute current values (e.g., \(I_1, I_2, I_3, I_4\)). It also covers dimensional analysis of electrical properties, focusing on combinations that yield force units.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_I%3A_Differential_Calculus/01%3A_Functions_and_GraphsIn this chapter, we review all the functions necessary to study calculus. We define polynomial, rational, trigonometric, exponential, and logarithmic functions. We review how to evaluate these functio...In this chapter, we review all the functions necessary to study calculus. We define polynomial, rational, trigonometric, exponential, and logarithmic functions. We review how to evaluate these functions, and we show the properties of their graphs. We provide examples of equations with terms involving these functions and illustrate the algebraic techniques necessary to solve them. In short, this chapter provides the foundation for the material to come.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_I%3A_Differential_Calculus/01%3A_Functions_and_Graphs/1.03%3A_Trigonometric_Functions/1.3E%3A_Exercises_for_Section_1.3This page contains exercises on angle conversions, trigonometric functions, triangle side calculations, and solving identities. It covers sine and cosine equations, including amplitude, period, and ph...This page contains exercises on angle conversions, trigonometric functions, triangle side calculations, and solving identities. It covers sine and cosine equations, including amplitude, period, and phase shifts, along with applications in geometry, angular speed, and natural phenomena models. Additionally, it features mathematical models predicting temperature and tide height with their respective amplitudes and periods, complemented by graphs illustrating their periodic variations.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/03%3A_Higher_order_linear_ODEs/3.08%3A_Mechanical_VibrationsLet us look at some applications of linear second order constant coefficient equations.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/01%3A_Systems_of_Equations/1.03%3A_Rank_and_Homogeneous_Systems/1.3E%3A_Exercises_for_Section_1.3This page discusses exercises on determining the rank of matrices and their implications for matrix equations. It features specific examples with ranks identified (e.g., 2 and 3) and explores the limi...This page discusses exercises on determining the rank of matrices and their implications for matrix equations. It features specific examples with ranks identified (e.g., 2 and 3) and explores the limitations of rank concerning matrix dimensions. The text also examines a hypothetical system of equations, emphasizing that some manipulations do not yield valid solutions.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/02%3A_Systems_of_Linear_Equations-_Geometry/2.08%3A_The_Rank_TheoremThis page explains the rank theorem, which connects a matrix's column space with its null space, asserting that the sum of rank (dimension of the column space) and nullity (dimension of the null space...This page explains the rank theorem, which connects a matrix's column space with its null space, asserting that the sum of rank (dimension of the column space) and nullity (dimension of the null space) equals the number of columns. It includes examples demonstrating how different ranks and nullities influence solution options in linear equations, emphasizing the theorem's importance in understanding the relationship between solution freedom and system properties without direct calculations.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/02%3A_Applications_of_First_Order_Equations/2.06%3A_Pursuit_CurvesThis page discusses pursuit curves involving a hawk chasing a sparrow on the y-axis. It derives equations that relate the hawk's interception path to both its and the sparrow's speeds. By solving a di...This page discusses pursuit curves involving a hawk chasing a sparrow on the y-axis. It derives equations that relate the hawk's interception path to both its and the sparrow's speeds. By solving a differential equation, the hawk's trajectory is determined, revealing that the hawk can successfully catch the sparrow if it travels faster than the sparrow.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/01%3A_Systems_of_Equations/1.06%3A_Application_to_Polynomial_InterpolationGiven \(n\) points in in \(\mathbb{R}^2\), we solve a linear system to find an appropriate degree polynomial that passes through them.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/04%3A_R/4.04%3A_Spanning_Sets_in_RBy 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.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/00%3A_Front_Matter/05%3A_8.1%3A_OverviewThis page is a textbook on linear algebra that integrates algebra and geometry, focusing on solving matrix equations and eigenvalue problems. It includes practical applications in various fields like ...This page is a textbook on linear algebra that integrates algebra and geometry, focusing on solving matrix equations and eigenvalue problems. It includes practical applications in various fields like engineering and biology, and discusses measurement errors in systems of equations. The text is structured for Math 1553 at Georgia Tech, featuring key concepts, algorithms, and interactive demos for enhanced learning.
