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7: Linear Transformations

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    58875

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    General linear transformations are introduced, motivated by many examples from geometry, matrix theory, and calculus. Then kernels and images are defined, the dimension theorem is proved, and isomorphisms are discussed. The chapter ends with an application to linear recurrences. A proof is included that the order of a differential equation (with constant coefficients) equals the dimension of the space of solutions

    • 7.0: Prelude to Linear Transformations
    • 7.1: Examples and Elementary Properties
      This page covers linear transformations \(T: V \to W\), outlining their defining properties, such as preserving vector addition and scalar multiplication, and the concept of linear operators. It details key attributes like the preservation of the zero vector and linear combinations, establishing criteria for equality of transformations based on their action on a spanning set.
    • 7.2: Kernel and Image of a Linear Transformation
      This page covers essential aspects of linear transformations, focusing on the kernel and image. The kernel identifies vectors mapping to zero, while the image reflects the outputs from these transformations. It discusses properties of one-to-one (trivial kernel) and onto (complete image) transformations, the Dimension Theorem linking dimensions (nullity + rank), and verifying polynomial evaluation mappings.
    • 7.3: Isomorphisms and Composition
      This page discusses the concept of isomorphisms in vector spaces, explaining that different representations can reflect the same underlying space. It covers the criteria for two finite-dimensional spaces to be isomorphic via linear transformations and bases. The significance of linear transformations, their compositions, and the existence of unique inverses under isomorphisms are emphasized.
    • 7.4: A Theorem about Differential Equations
      This page covers the significance of differential equations in science, introducing linear differential equations with constant coefficients and their solution sets as vector spaces. It explores complex-valued functions and establishes relationships between dimensions of solution spaces. The properties of linear operators on vector spaces are detailed, including key functions and differentiation rules.
    • 7.5: More on Linear Recurrences
      This page covers linear recurrences through the lens of vector spaces and linear transformations, establishing the representation of sequences as vectors. It discusses the linearity, injectiveness, and surjectiveness of transformation \(T\) on these sequences, while introducing the Vandermonde matrix and the shift operator.

    Thumbnail: 3D basis transformation. (Public Domain; Maschen via Wikipedia)

    This page titled 7: Linear Transformations is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by W. Keith Nicholson via source content that was edited to the style and standards of the LibreTexts platform.