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- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/04%3A_R/4.06%3A_Subspaces_and_BasesThe goal of this section is to develop an understanding of a subspace of \(\mathbb{R}^n\).
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/02%3A_Systems_of_Linear_Equations-_Geometry/2.02%3A_Vector_Equations_and_SpansThis page examines the relationships between systems of linear equations and vector equations, focusing on the span of vectors and its significance. It defines the span as all linear combinations of g...This page examines the relationships between systems of linear equations and vector equations, focusing on the span of vectors and its significance. It defines the span as all linear combinations of given vectors and explains that a vector is in the span if the corresponding linear equation is consistent.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/04%3A_R/4.09%3A_Gram-Schmidt_ProcessThe Gram-Schmidt process is an algorithm to transform a set of vectors into an orthonormal set spanning the same subspace, that is generating the same collection of linear combinations.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/02%3A_Systems_of_Linear_Equations-_Geometry/2.01%3A_VectorsThis page discusses the fundamental concepts of vectors in \(\mathbb{R}^n\), including their algebraic and geometric interpretations, addition, subtraction, and scalar multiplication. It highlights th...This page discusses the fundamental concepts of vectors in \(\mathbb{R}^n\), including their algebraic and geometric interpretations, addition, subtraction, and scalar multiplication. It highlights the parallelogram law, linear combinations of vectors, and their real-world applications, especially in physical quantities like velocity.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/07%3A_Vector_Spaces/7.08%3A_Linear_TransformationsIn this section we discuss the definition of a linear transformation in the context of vector spaces.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/04%3A_R/4.04%3A_Spanning_Sets_in_R/4.4.E%3A_Exercise_for_Section_4.4This page outlines exercises focused on vector spans, requiring the identification of minimal sets of vectors that span the same space, checking vector inclusions, and demonstrating linear combination...This page outlines exercises focused on vector spans, requiring the identification of minimal sets of vectors that span the same space, checking vector inclusions, and demonstrating linear combinations. It includes theoretical components about spans, emphasizing that they always include the zero vector.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/03%3A_Higher_order_linear_ODEs/3.01%3A_Second_order_linear_ODEsThis page covers second-order linear differential equations, addressing both homogeneous and nonhomogeneous types. Key concepts include the general form of these equations, the superposition principle...This page covers second-order linear differential equations, addressing both homogeneous and nonhomogeneous types. Key concepts include the general form of these equations, the superposition principle, and the uniqueness of solutions. Techniques like reduction of order for finding second solutions are detailed, illustrated with examples such as \(y_1 = x\). The text emphasizes understanding methods over memorizing formulas and sets the stage for tackling nonhomogeneous equations.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/07%3A_Vector_Spaces/7.09%3A_IsomorphismsThis page explores linear transformations in vector spaces, focusing on one-to-one and onto transformations and their role in defining isomorphisms. It establishes that a linear transformation \(T\) i...This page explores linear transformations in vector spaces, focusing on one-to-one and onto transformations and their role in defining isomorphisms. It establishes that a linear transformation \(T\) is an isomorphism if it is both one-to-one and onto, retaining linear independence of bases.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/04%3A_RThis page details vector concepts, covering operations like dot and cross products, lines and planes in \(\mathbb{R}^3\), and spanning sets. It discusses linear independence, matrix spaces (row, colum...This page details vector concepts, covering operations like dot and cross products, lines and planes in \(\mathbb{R}^3\), and spanning sets. It discusses linear independence, matrix spaces (row, column, null), orthogonal vectors/matrices, the Gram-Schmidt process for orthonormal sets, orthogonal projections, and least squares approximation. The page includes exercises for practice at the end of each section.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/05%3A_Linear_Transformations/5.03%3A_Properties_of_Linear_TransformationsLet \(T: \mathbb{R}^n \mapsto \mathbb{R}^m\) be a linear transformation. Then there are some important properties of \(T\) which will be examined in this section.
