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About 35 results
  • 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.
  • https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/A_First_Course_in_Linear_Algebra_(Kuttler)/09%3A_Vector_Spaces/9.03%3A_Linear_Independence
    In this section, we will again explore concepts introduced earlier in terms of Rn and extend them to apply to abstract vector spaces.
  • 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.
  • 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.
  • https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/07%3A_Vector_Spaces/7.04%3A_Linear_Independence
    In this section, we will again explore concepts introduced earlier in terms of Rn and extend them to apply to abstract vector spaces.
  • https://math.libretexts.org/Bookshelves/Linear_Algebra/A_First_Course_in_Linear_Algebra_(Kuttler)/09%3A_Vector_Spaces/9.03%3A_Linear_Independence
    In this section, we will again explore concepts introduced earlier in terms of Rn and extend them to apply to abstract vector spaces.
  • 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.
  • https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/06%3A_Systems_of_ODEs/6.02%3A_Linear_systems_of_ODEs/6.2E%3A_Exercises_for_Section_6.2
    This page details exercises focused on systems of differential equations, including verifying solutions, checking the linear independence of vectors, and matrix representations. Key tasks involve find...This page details exercises focused on systems of differential equations, including verifying solutions, checking the linear independence of vectors, and matrix representations. Key tasks involve finding general solutions, validating specific solutions for given matrices, and deriving forms with mathematical justification. The exercises explore various systems and functions, necessitating in-depth understanding and presentation of results linked to independence and matrix notation.
  • https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/03%3A_Higher_order_linear_ODEs
    In this chapter, we go a little further and look at second-order equations, which are equations containing second derivatives of the dependent variable. The solution methods we examine are different f...In this chapter, we go a little further and look at second-order equations, which are equations containing second derivatives of the dependent variable. The solution methods we examine are different from those discussed earlier, and the solutions tend to involve trigonometric functions as well as exponential functions. Here we concentrate primarily on second-order equations with constant coefficients.
  • https://math.libretexts.org/Courses/Community_College_of_Denver/MAT_2562_Differential_Equations_with_Linear_Algebra/13%3A_Vector_Spaces/13.03%3A_Linear_Independence
    In this section, we will again explore concepts introduced earlier in terms of Rn and extend them to apply to abstract vector spaces.
  • https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/07%3A_Vector_Spaces/7.04%3A_Linear_Independence/7.4E%3A_Exercises_for_Section_7.4
    This page discusses exercises on linear independence of vectors and polynomials. It concludes that a specific set of polynomials is linearly independent, while another set is determined to be linearly...This page discusses exercises on linear independence of vectors and polynomials. It concludes that a specific set of polynomials is linearly independent, while another set is determined to be linearly dependent with a provided linear combination. Additional exercises examine conditions for independence in polynomial collections and propose proof methods for assessing the independence of three polynomials in a given space.

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