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5.8: The General Solution of a Linear System
https://math.libretexts.org/Courses/Reedley_College/Differential_Equations_and_Linear_Algebra_(Zook)/05%3A_Linear_Transformations/5.08%3A_The_General_Solution_of_a_Linear_System
It turns out that we can use linear transformations to solve linear systems of equations.
3.2: Null Space
https://math.libretexts.org/Bookshelves/Linear_Algebra/Matrix_Analysis_(Cox)/03%3A_The_Fundamental_Subspaces/3.02%3A_Null_Space
The null space of an $m$ -by- $n$ matrix $A$ is the collection of those vectors in $\mathbb{R}^{n}$ that $A$ maps to the zero vector in $\mathbb{R}^m$ . If we use the column space to determin...The null space of an $m$ -by- $n$ matrix $A$ is the collection of those vectors in $\mathbb{R}^{n}$ that $A$ maps to the zero vector in $\mathbb{R}^m$ . If we use the column space to determine the existence of a solution $\textbf{x}$ to the equation $A \textbf{⁢x} = b$ . The hard and fast rule is that a solution $\textbf{x}$ is unique if and only if the null space of $A$ is empty.
5.1: Eigenvalues and Eigenvectors
https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/05%3A_Eigenvalues_and_Eigenvectors/5.01%3A_Eigenvalues_and_Eigenvectors
This page explains eigenvalues and eigenvectors in linear algebra, detailing their definitions, significance, and processes for finding them. It discusses how eigenvectors result from matrix transform...This page explains eigenvalues and eigenvectors in linear algebra, detailing their definitions, significance, and processes for finding them. It discusses how eigenvectors result from matrix transformations and the linear independence of distinct eigenvectors. The text covers specific examples, including eigenvalue analysis for specific matrices and the conditions for eigenvalues, including zero.
2.7: Basis and Dimension
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.
5.9: The General Solution of a Linear System
https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/05%3A_Linear_Transformations/5.09%3A_The_General_Solution_of_a_Linear_System
In this section we see how to use linear transformations to solve linear systems of equations.
5.9: The General Solution of a Linear System
https://math.libretexts.org/Courses/Coastline_College/Math_C285%3A_Linear_Algebra_and_Diffrential_Equations_(Tran)/05%3A_Linear_Transformations/5.09%3A_The_General_Solution_of_a_Linear_System
It turns out that we can use linear transformations to solve linear systems of equations.
3.3: The Null and Column Spaces- An Example
https://math.libretexts.org/Bookshelves/Linear_Algebra/Matrix_Analysis_(Cox)/03%3A_The_Fundamental_Subspaces/3.03%3A_The_Null_and_Column_Spaces-_An_Example
In our example there are six of each and, again on account of the staircase nature, the pivot columns are the linearly independent columns of $A_{red}$ One now asks how this might help us distinguis...In our example there are six of each and, again on account of the staircase nature, the pivot columns are the linearly independent columns of $A_{red}$ One now asks how this might help us distinguish the independent columns of A. As $x_{6}$ and $x_{8}$ range over all real numbers, the $\textbf{x}$ above traces out a plane in $\mathbb{R}^8$ This plane is precisely the null space of A and Equation describes a generic element as the linear combination of two basis vectors.
6.2: Orthogonal Complements
https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/06%3A_Orthogonality/6.02%3A_Orthogonal_Complements
This page explores orthogonal complements in linear algebra, defining them as vectors orthogonal to a subspace $W$ in $\mathbb{R}^n$ . It details properties, computation methods (such as using RREF...This page explores orthogonal complements in linear algebra, defining them as vectors orthogonal to a subspace $W$ in $\mathbb{R}^n$ . It details properties, computation methods (such as using RREF), and visual representations in $\mathbb{R}^2$ and $\mathbb{R}^3$ . Key concepts include the relationship between a subspace and its double orthogonal complement, the equality of row and column ranks of matrices, and the significance of dimensions in relation to null spaces.
4.7: Row, Column and Null Spaces
https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/04%3A_R/4.07%3A_Row_Column_and_Null_Spaces
This section discusses the Row, Column, and Null Spaces of a matrix, focusing on their definitions, properties, and computational methods.
4.9: The General Solution of a Linear System
https://math.libretexts.org/Courses/Mission_College/MAT_04C_Linear_Algebra_(Kravets)/04%3A_Linear_Transformations/4.09%3A_The_General_Solution_of_a_Linear_System
It turns out that we can use linear transformations to solve linear systems of equations.
2.6: Subspaces
https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/02%3A_Systems_of_Linear_Equations-_Geometry/2.06%3A_Subspaces
This page defines subspaces in $\mathbb{R}^n$ and outlines criteria for a subset to qualify as a subspace, including non-emptiness and closure under addition and scalar multiplication. It offers exa...This page defines subspaces in $\mathbb{R}^n$ and outlines criteria for a subset to qualify as a subspace, including non-emptiness and closure under addition and scalar multiplication. It offers examples of valid and invalid subspaces, discusses conditions specific to $\mathbb{R}^2$ , and explains spanning sets of subspaces, particularly the column and null spaces of matrices.

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