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- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/03%3A_Linear_Transformations_and_Matrix_Algebra/3.02%3A_One-to-one_and_Onto_TransformationsThis page discusses the concepts of one-to-one and onto transformations in linear algebra, focusing on matrix transformations. It defines one-to-one as each output having at most one input and outline...This page discusses the concepts of one-to-one and onto transformations in linear algebra, focusing on matrix transformations. It defines one-to-one as each output having at most one input and outlines examples and theorems related to this property. The text emphasizes that a transformation is onto if every output corresponds to some input.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/01%3A_Systems_of_Equations/1.01%3A_Systems_of_Linear_EquationsThis page discusses methods for solving systems of equations with two or three variables, covering unique, infinite, and no solutions. It emphasizes the significance of graphical representations, inte...This page discusses methods for solving systems of equations with two or three variables, covering unique, infinite, and no solutions. It emphasizes the significance of graphical representations, intersections of lines and planes, and the complexities introduced by additional variables. The page addresses consistent versus inconsistent systems, homogeneous systems, and the application of elementary operations that preserve solution sets.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/01%3A_Systems_of_Linear_Equations-_Algebra/1.02%3A_Row_ReductionThis page introduces the elimination method for solving systems of linear equations using augmented matrices and row operations. It defines applicable operations, including scaling, replacement, and s...This page introduces the elimination method for solving systems of linear equations using augmented matrices and row operations. It defines applicable operations, including scaling, replacement, and swapping, and discusses achieving row echelon and reduced row echelon forms. The process includes identifying unique, inconsistent, and infinite solutions through examples, emphasizing the significance of pivot positions.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/01%3A_Systems_of_Linear_Equations-_Algebra/1.01%3A_Systems_of_Linear_EquationsThis page introduces \(\mathbb{R}^n\) as the set of ordered \(n\)-tuples of real numbers for labeling geometric points, focusing on linear equations' structure, consistency, and solutions. It discusse...This page introduces \(\mathbb{R}^n\) as the set of ordered \(n\)-tuples of real numbers for labeling geometric points, focusing on linear equations' structure, consistency, and solutions. It discusses the geometric interpretation of solutions in n-dimensional space, illustrating how linear equations define lines or planes.
- 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.