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20.2: Subspaces
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Abstract_Algebra%3A_Theory_and_Applications_(Judson)/20%3A_Vector_Spaces/20.02%3A_Subspaces
Let $V$ be a vector space over a field $F\text{,}$ and $W$ a subset of $V\text{.}$ Then $W$ is a subspace of $V$ if it is closed under vector addition and scalar multiplication; that is, i...Let $V$ be a vector space over a field $F\text{,}$ and $W$ a subset of $V\text{.}$ Then $W$ is a subspace of $V$ if it is closed under vector addition and scalar multiplication; that is, if $u, v \in W$ and $\alpha \in F\text{,}$ it will always be the case that $u + v$ and $\alpha v$ are also in $W\text{.}$
7.7: Sums and Intersections
https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/07%3A_Vector_Spaces/7.07%3A_Sums_and_Intersections
In this section we discuss sum and intersection of two subspaces.
5.5E: Exercises for Section 5.5
https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/05%3A_Linear_Transformations/5.05%3A_One-to-One_and_Onto_Transformations/5.5E%3A_Exercises_for_Section_5.5
This page features exercises on linear transformations and their matrix representations, focusing on properties such as injectivity and surjectivity. It includes tasks to analyze various matrix sizes ...This page features exercises on linear transformations and their matrix representations, focusing on properties such as injectivity and surjectivity. It includes tasks to analyze various matrix sizes and examines the relationship between rank, linear independence, and the properties of transformations. Overall, the content emphasizes essential linear algebra concepts related to the effects of transformations on vector spaces.
4.4: Spanning Sets in Rⁿ
https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/04%3A_R/4.04%3A_Spanning_Sets_in_R
By generating all linear combinations of a set of vectors one can obtain various subsets of $\mathbb{R}^{n}$ which we call subspaces. For example what set of vectors in $\mathbb{R}^{3}$ generate t...By generating all linear combinations of a set of vectors one can obtain various subsets of $\mathbb{R}^{n}$ which we call subspaces. For example what set of vectors in $\mathbb{R}^{3}$ generate the $XY$ -plane? What is the smallest such set of vectors can you find? The tools of spanning, linear independence and basis are exactly what is needed to answer these and similar questions and are the focus of this section.
7.10: The Kernel and Image of a Linear Map
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
Here we consider the case where the linear map is not necessarily an isomorphism. First here is a definition of what is meant by the image and kernel of a linear transformation.
5.6: Isomorphisms
https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/05%3A_Linear_Transformations/5.06%3A_Isomorphisms
In this section we will consider linear transformations that are one to one and onto. Such linear transformations are called isomorphisms.
5.3E: Exercises for Section 5.3
https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/05%3A_Linear_Transformations/5.03%3A_Properties_of_Linear_Transformations/5.3E%3A_Exercises_for_Section_5.3
This page contains exercises on linear transformations, emphasizing that they map the zero vector to itself. It includes computations with matrices, composition of transformations, evaluations at spec...This page contains exercises on linear transformations, emphasizing that they map the zero vector to itself. It includes computations with matrices, composition of transformations, evaluations at specific vectors, and finding inverse matrices. Additionally, there is a theoretical exercise related to the linear independence of sets within subspaces of $\mathbb{R}^n$ . The tasks encompass both practical calculations and proofs.
2: Systems of Linear Equations- Geometry
https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/02%3A_Systems_of_Linear_Equations-_Geometry
This page covers expressing systems of linear equations as matrix equations $Ax=b$ , exploring solution sets and consistency conditions. It utilizes a geometric perspective to analyze the solution se...This page covers expressing systems of linear equations as matrix equations $Ax=b$ , exploring solution sets and consistency conditions. It utilizes a geometric perspective to analyze the solution set and column space of matrix $A$ . Key concepts include vectors, spans, linear independence, subspaces, and dimension, all aimed at enhancing geometric understanding. The rank theorem further clarifies the relationships among these ideas.
7: Vector Spaces
https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/07%3A_Vector_Spaces
This page defines vector spaces and describes their properties, including operations, spanning sets, linear independence, and subspaces. It covers bases, subspace operations, linear transformations, a...This page defines vector spaces and describes their properties, including operations, spanning sets, linear independence, and subspaces. It covers bases, subspace operations, linear transformations, and the concepts of image and kernel. The text also discusses the matrix representation of linear transformations and introduces inner product spaces, which apply geometric concepts of length and orthogonality to general vector spaces.
5.7: The Kernel and Image of A Linear Map
https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/05%3A_Linear_Transformations/5.07%3A_The_Kernel_and_Image_of_A_Linear_Map
In this section we will consider two important subspaces associated with a linear transformation: its kernel and its image.
4.6.E: Exercise for Section 4.6
https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/04%3A_R/4.06%3A_Subspaces_and_Bases/4.6.E%3A_Exercise_for_Section_4.6
This page discusses exercises in linear algebra focused on subspaces in $\mathbb{R}^3$ and $\mathbb{R}^4$ . It analyzes spans of vectors to determine their dimensions and bases, assesses conditions...This page discusses exercises in linear algebra focused on subspaces in $\mathbb{R}^3$ and $\mathbb{R}^4$ . It analyzes spans of vectors to determine their dimensions and bases, assesses conditions for specific sets to qualify as subspaces by examining closure under addition and scalar multiplication, and provides counterexamples. Additionally, it explores properties of linearly independent sets, particularly the influence of the number of vectors in $\mathbb{R}^5$ .

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