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- https://math.libretexts.org/Courses/Los_Angeles_City_College/Math_230-Mathematics_for_Liberal_Arts_Students/02%3A_Matrices/2.01%3A_Introduction_to_MatricesA matrix is a 2 dimensional array of numbers arranged in rows and columns. Matrices provide a method of organizing, storing, and working with mathematical information. Matrices have an abundance of ap...A matrix is a 2 dimensional array of numbers arranged in rows and columns. Matrices provide a method of organizing, storing, and working with mathematical information. Matrices have an abundance of applications and use in the real world. Matrices provide a useful tool for working with models based on systems of linear equations. We’ll use matrices in sections 2.2, 2.3, and 2.4 to solve systems of linear equations with several variables in this chapter.
- 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_DimensionThis 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.
- 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/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/04%3A_R/4.06%3A_Subspaces_and_BasesThe goal of this section is to develop an understanding of a subspace of \(\mathbb{R}^n\).
- https://math.libretexts.org/Workbench/1250_Draft_4/07%3A_Matrices/7.03%3A_Introduction_to_MatricesA matrix is a 2 dimensional array of numbers arranged in rows and columns. Matrices provide a method of organizing, storing, and working with mathematical information. Matrices have an abundance of ap...A matrix is a 2 dimensional array of numbers arranged in rows and columns. Matrices provide a method of organizing, storing, and working with mathematical information. Matrices have an abundance of applications and use in the real world. Matrices provide a useful tool for working with models based on systems of linear equations. We’ll use matrices in sections 2.2, 2.3, and 2.4 to solve systems of linear equations with several variables in this chapter.
- 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/Bookshelves/Applied_Mathematics/Applied_Finite_Mathematics_(Sekhon_and_Bloom)/02%3A_Matrices/2.01%3A_Introduction_to_MatricesA matrix is a 2 dimensional array of numbers arranged in rows and columns. Matrices provide a method of organizing, storing, and working with mathematical information. Matrices have an abundance of ap...A matrix is a 2 dimensional array of numbers arranged in rows and columns. Matrices provide a method of organizing, storing, and working with mathematical information. Matrices have an abundance of applications and use in the real world. Matrices provide a useful tool for working with models based on systems of linear equations. We’ll use matrices in sections 2.2, 2.3, and 2.4 to solve systems of linear equations with several variables in this chapter.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/07%3A_Vector_Spaces/7.11%3A_The_Matrix_of_a_Linear_TransformationYou may recall from Rn that the matrix of a linear transformation depends on the bases chosen. This concept is explored in this section, where the linear transformation now maps from one arbitrary v...You may recall from Rn that the matrix of a linear transformation depends on the bases chosen. This concept is explored in this section, where the linear transformation now maps from one arbitrary vector space to another.
- https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/MAT-204%3A_Differential_Equations_for_Science_(Lebl_and_Trench)/11%3A_Appendix_A-_Linear_Algebra/11.04%3A_A.4-_Subspaces_Dimension_and_The_KernelIt is the 3-dimensional space \[\text{column space of $L$} = \operatorname{span} \left\{ \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} , \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} , \begin{bmatrix} 0 \\ 0 ...It is the 3-dimensional space \[\text{column space of $L$} = \operatorname{span} \left\{ \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} , \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} , \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \right\} = {\mathbb{R}}^3 . \nonumber \] The row space is the 3-dimensional space \[\text{row space of $L$} = \operatorname{span} \left\{ \begin{bmatrix} 1 & 2 & 0 & 0 & 3 \end{bmatrix} , \begin{bmatrix} 0 & 0 & 1 & 0 & 4 \end{bmatrix} , \begin{bmatrix} 0 & 0 & 0 & 1 & 5 \end{bma…
- 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/07%3A_Vector_Spaces/7.11%3A_The_Matrix_of_a_Linear_Transformation/7.11E%3A_Exercises_for_Section_7.11This page presents exercises on linear transformations and matrix representations across vector spaces like \(\mathbb{R}^2\), \(\mathbb{P}_2\), and \(M_{22}\). It includes finding coordinate vectors, ...This page presents exercises on linear transformations and matrix representations across vector spaces like \(\mathbb{R}^2\), \(\mathbb{P}_2\), and \(M_{22}\). It includes finding coordinate vectors, matrices under specified bases, and exploring the kernel and image of transformations. Exercises feature transformations, such as differentiating polynomials and mapping \(2 \times 2\) matrices to \(\mathbb{R}^2\), complete with specified bases and matrix representations.
