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- 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/Linear_Algebra%3A_A_First_Course/04%3A_R/4.07%3A_Row_Column_and_Null_SpacesThis section discusses the Row, Column, and Null Spaces of a matrix, focusing on their definitions, properties, and computational methods.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/03%3A_Linear_Transformations_and_Matrix_Algebra/3.06%3A_The_Invertible_Matrix_TheoremThis page explores the Invertible Matrix Theorem, detailing equivalent conditions for a square matrix \(A\) to be invertible, such as having \(n\) pivots and unique solutions for \(Ax=b\). It includes...This page explores the Invertible Matrix Theorem, detailing equivalent conditions for a square matrix \(A\) to be invertible, such as having \(n\) pivots and unique solutions for \(Ax=b\). It includes proofs and examples, emphasizes the theorem's importance, and presents a corollary linking inverses to invertibility.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/07%3A_Vector_Spaces/7.10%3A_The_Kernel_and_Image_of_a_Linear_Map/7.10E%3A_Exercises_for_Section_7.10This page contains exercises on linear transformations between vector spaces, focusing on determining kernels and images, proving linearity, finding bases, and identifying rank and nullity. Key tasks ...This page contains exercises on linear transformations between vector spaces, focusing on determining kernels and images, proving linearity, finding bases, and identifying rank and nullity. Key tasks involve transformations between \(\mathbb{R}^2\) and \(\mathbb{R}^3\) to \(\mathbb{R}^2\), along with polynomial transformations. The set concludes with exploration of vector space isomorphisms in \(\mathbb{R}^6\).