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- https://math.libretexts.org/Bookshelves/Linear_Algebra/Linear_Algebra_with_Applications_(Nicholson)/06%3A_Vector_Spaces/6.01%3A_Examples_and_Basic_PropertiesA vector space consists of a nonempty set V of objects (called vectors) that can be added, that can be multiplied by a real number (called a scalar in this context), and for which certain axioms ho...A vector space consists of a nonempty set V of objects (called vectors) that can be added, that can be multiplied by a real number (called a scalar in this context), and for which certain axioms hold.
- https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/A_First_Course_in_Linear_Algebra_(Kuttler)/09%3A_Vector_Spaces/9.01%3A_Algebraic_ConsiderationsIn this section we consider the idea of an abstract vector space.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/A_First_Course_in_Linear_Algebra_(Kuttler)/09%3A_Vector_Spaces/9.01%3A_Algebraic_ConsiderationsIn this section we consider the idea of an abstract vector space.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/07%3A_Vector_Spaces/7.05%3A_SubspacesIn 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.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/07%3A_Vector_Spaces/7.07%3A_Sums_and_IntersectionsIn this section we discuss sum and intersection of two subspaces.
- https://math.libretexts.org/Workbench/Math_26%3A_Introduction_to_Linear_Algebra/04%3A_Vector_Spaces/4.01%3A_General_Vector_Spaces_and_SubspacesIn this section we consider the idea of an abstract vector space.
- https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/A_First_Journey_Through_Linear_Algebra/05%3A_Vector_Spaces/5.01%3A_Algebraic_ConsiderationsIn this section we consider the idea of an abstract vector space.
- https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/A_First_Journey_Through_Linear_Algebra/05%3A_Vector_Spaces/5.11%3A_Supplementary_Notes_-_A_More_In-Depth_Look_at_Vector_Spaces/5.11.01%3A_Vector_Spaces/5.11.1.01%3A_Examples_and_Basic_PropertiesA vector space consists of a nonempty set V of objects (called vectors) that can be added, that can be multiplied by a real number (called a scalar in this context), and for which certain axioms ho...A vector space consists of a nonempty set V of objects (called vectors) that can be added, that can be multiplied by a real number (called a scalar in this context), and for which certain axioms hold.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Matrix_Analysis_(Cox)/03%3A_The_Fundamental_Subspaces/3.07%3A_Supplements_-_Vector_Space\((c_{1}c_{2}) \textbf{x} = c_{1}(c_{2} \textbf{x})\) for each \(\textbf{x}\) in \(V\) and \(c_{1}\) and \(c_{2}\) in \(\mathbb{C}\). \(c(\textbf{x}+\textbf{y}) = c\textbf{x}+c\textbf{y}\) for each \(...\((c_{1}c_{2}) \textbf{x} = c_{1}(c_{2} \textbf{x})\) for each \(\textbf{x}\) in \(V\) and \(c_{1}\) and \(c_{2}\) in \(\mathbb{C}\). \(c(\textbf{x}+\textbf{y}) = c\textbf{x}+c\textbf{y}\) for each \(\textbf{x}\) and \(\textbf{y}\) in \(V\) and c in \(\mathbb{C}\). \((c_{1}+c_{2}) \textbf{x} = c_{1} \textbf{x}+c_{2} \textbf{x}\) for each \(\textbf{x}\) in \(V\) and \(c_{1}\) and \(c_{2}\) in \(\mathbb{C}\).
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/02%3A_Systems_of_Linear_Equations-_Geometry/2.04%3A_Solution_SetsThis page discusses homogeneous and inhomogeneous linear systems, focusing on equations \(Ax=0\) and \(Ax=b\). It defines homogeneous systems as having zero constants, always including the trivial sol...This page discusses homogeneous and inhomogeneous linear systems, focusing on equations \(Ax=0\) and \(Ax=b\). It defines homogeneous systems as having zero constants, always including the trivial solution \(x=0\). Solutions are expressed as spans of vectors, with their dimensions linked to free variables. For consistent equations \(Ax=b\), solutions can be formulated as a particular solution plus the homogeneous solution set.
- 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_5.6This 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.
