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1.5: Images and Inverses
https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Transition_to_Higher_Mathematics_(Dumas_and_McCarthy)/01%3A_New_Page/1.5%3A_Images_and_Inverses
The identity function on $X$ , id $\left.\right|_{X}: X \mapsto X$ , is the function defined by $\left.\operatorname{id}\right|_{X}(x)=x .$ If $f: X \rightarrow Y$ is a bijection, then \(f^{-1}\...The identity function on $X$ , id $\left.\right|_{X}: X \mapsto X$ , is the function defined by $\left.\operatorname{id}\right|_{X}(x)=x .$ If $f: X \rightarrow Y$ is a bijection, then $f^{-1}$ is the unique function such that $f^{-1} \circ f=\left.\operatorname{id}\right|_{X}$ and $f \circ f^{-1}=\left.\mathrm{id}\right|_{Y} .$ Because $f(x)=x^{2}$ is not an injection, it has no inverse, even after restricting the codomain to be the range.
5.6E: Exercises for Section 5.6
https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/05%3A_Linear_Transformations/5.06%3A_Isomorphisms/5.6E%3A_Exercises_for_Section_5.6
This page contains exercises on linear transformations and isomorphisms in vector spaces, focusing on defining transformations from $\mathbb{R}^3$ and $\mathbb{R}^2$ . It covers properties of isomo...This page contains exercises on linear transformations and isomorphisms in vector spaces, focusing on defining transformations from $\mathbb{R}^3$ and $\mathbb{R}^2$ . It covers properties of isomorphisms, proving conditions for transformations, exploring matrix representations, and finding inverses. The content also discusses constructing matrices that uphold the structure of transformations and their inverses, particularly regarding spans of vectors in higher dimensions.
9.2: Building Subspaces
https://math.libretexts.org/Bookshelves/Linear_Algebra/Map%3A_Linear_Algebra_(Waldron_Cherney_and_Denton)/09%3A_Subspaces_and_Spanning_Sets/9.02%3A_Building_Subspaces
The vector $\begin{pmatrix}2 \\ 3 \\ 0\end{pmatrix}$ is in $span(S),$ because $\begin{pmatrix}2\\3\\0\end{pmatrix}=\begin{pmatrix}2\\0\\0\end{pmatrix}+3\begin{pmatrix}0\\1\\0\end{pmatrix}.$ Simi...The vector $\begin{pmatrix}2 \\ 3 \\ 0\end{pmatrix}$ is in $span(S),$ because $\begin{pmatrix}2\\3\\0\end{pmatrix}=\begin{pmatrix}2\\0\\0\end{pmatrix}+3\begin{pmatrix}0\\1\\0\end{pmatrix}.$ Similarly, the vector $\begin{pmatrix}-12 \\ 17.5 \\ 0\end{pmatrix}$ is in $span(S),$ because $\begin{pmatrix}-12\\17.5\\0\end{pmatrix}=\begin{pmatrix}-12\\0\\0\end{pmatrix}+17.5\begin{pmatrix}0\\1\\0\end{pmatrix}.$
7.11E: Exercises for Section 7.11
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_7.11
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, ...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.
1.2: Functions
https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/First-Semester_Abstract_Algebra%3A_A_Structural_Approach_(Sklar)/01%3A_Preliminaries/1.02%3A_Functions
You have probably encountered functions before. In introductory calculus, for instance, you typically deal with functions from real numbers to real numbers (e.g., the function f(x) = x^2). More gene...You have probably encountered functions before. In introductory calculus, for instance, you typically deal with functions from real numbers to real numbers (e.g., the function f(x) = x^2). More generally, functions “send” elements of one set to elements of another set; these sets may or may not be sets of real numbers. We provide below a “good enough for government work” definition of a function.
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.
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.
3.1: Basic Transformations of Complex Numbers
https://math.libretexts.org/Bookshelves/Geometry/Geometry_with_an_Introduction_to_Cosmic_Topology_(Hitchman)/03%3A_Transformations/3.01%3A_Basic_Transformations_of_Complex_Numbers
In this section, we develop the following basic transformations of the plane, as well as some of their important features.
14.2: Building Subspaces
https://math.libretexts.org/Under_Construction/Purgatory/Differential_Equations_and_Linear_Algebra_(Zook)/14%3A_Subspaces_and_Spanning_Sets/14.02%3A_Building_Subspaces
The vector $\begin{pmatrix}2 \\ 3 \\ 0\end{pmatrix}$ is in $span(S),$ because $\begin{pmatrix}2\\3\\0\end{pmatrix}=\begin{pmatrix}2\\0\\0\end{pmatrix}+3\begin{pmatrix}0\\1\\0\end{pmatrix}.$ Simi...The vector $\begin{pmatrix}2 \\ 3 \\ 0\end{pmatrix}$ is in $span(S),$ because $\begin{pmatrix}2\\3\\0\end{pmatrix}=\begin{pmatrix}2\\0\\0\end{pmatrix}+3\begin{pmatrix}0\\1\\0\end{pmatrix}.$ Similarly, the vector $\begin{pmatrix}-12 \\ 17.5 \\ 0\end{pmatrix}$ is in $span(S),$ because $\begin{pmatrix}-12\\17.5\\0\end{pmatrix}=\begin{pmatrix}-12\\0\\0\end{pmatrix}+17.5\begin{pmatrix}0\\1\\0\end{pmatrix}.$
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.
5.7E: Exercises for Section 5.7
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/5.7E%3A_Exercises_for_Section_5.7
This page covers exercises on linear transformations and vector spaces, focusing on finding bases for spans, kernels, and images. It highlights key results like dimension identification, one-to-one tr...This page covers exercises on linear transformations and vector spaces, focusing on finding bases for spans, kernels, and images. It highlights key results like dimension identification, one-to-one transformations, and the distinctions between independent and dependent vector sets.

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