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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.
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.
3.1: Column Space
https://math.libretexts.org/Bookshelves/Linear_Algebra/Matrix_Analysis_(Cox)/03%3A_The_Fundamental_Subspaces/3.01%3A_Column_Space
Namely, the multiplication of the n-by-1 vector $x$ by the m-by-n matrix $A$ produces a linear combination of the columns of A. The column space of the m-by-n matrix $S$ is simply the span of th...Namely, the multiplication of the n-by-1 vector $x$ by the m-by-n matrix $A$ produces a linear combination of the columns of A. The column space of the m-by-n matrix $S$ is simply the span of the its columns, i.e. $Ra(S) \equiv \{Sx | x \in \mathbb{R}^{n}\}$ subspace of $\mathcal{R}^{m}$ stands for range in this context.The notation $R_{a}$ stands for range in this context.
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.
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.
4.7.E: Exercise for Section 4.7
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/4.7.E%3A_Exercise_for_Section_4.7
This page presents exercises on matrices, emphasizing the calculation of bases for row, column, and null spaces, alongside ranks and nullities. It validates the Rank-Nullity Theorem and explores kerne...This page presents exercises on matrices, emphasizing the calculation of bases for row, column, and null spaces, alongside ranks and nullities. It validates the Rank-Nullity Theorem and explores kernel spaces as subspaces of $\mathbb{R}^n$ . Key topics include linearly independent rows, pivot columns, and methods for solving linear algebra problems.
5.10.1.3: The Null and Column Spaces- An Example
https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/A_First_Journey_Through_Linear_Algebra/05%3A_Vector_Spaces/5.10%3A_Supplementary_Notes_-_The_Fundamental_Subspaces/5.10.01%3A_The_Fundamental_Subspaces/5.10.1.03%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.
3.6: The Invertible Matrix Theorem
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_Theorem
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...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.
5.10.1.1: Column Space
https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/A_First_Journey_Through_Linear_Algebra/05%3A_Vector_Spaces/5.10%3A_Supplementary_Notes_-_The_Fundamental_Subspaces/5.10.01%3A_The_Fundamental_Subspaces/5.10.1.01%3A_Column_Space
Namely, the multiplication of the n-by-1 vector $x$ by the m-by-n matrix $A$ produces a linear combination of the columns of A. The column space of the m-by-n matrix $S$ is simply the span of th...Namely, the multiplication of the n-by-1 vector $x$ by the m-by-n matrix $A$ produces a linear combination of the columns of A. The column space of the m-by-n matrix $S$ is simply the span of the its columns, i.e. $Ra(S) \equiv \{Sx | x \in \mathbb{R}^{n}\}$ subspace of $\mathcal{R}^{m}$ stands for range in this context.The notation $R_{a}$ stands for range in this context.

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