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    Graph of Linear Transformations over a Field

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    29.232-2019.01 Yasmine Ahmed El-Ashi.pdf (1.787Mb)
    Date
    2019-06
    Author
    El-Ashi, Yasmine Ahmed
    Advisor(s)
    Badawi, Ayman
    Type
    Thesis
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    Description
    A Master of Science thesis in Mathematics by Yasmine Ahmed El-Ashi entitled, “Graph of Linear Transformations over a Field”, submitted in June 2019. Thesis advisor is Dr. Ayman Badawi. Soft and hard copy available.
    Abstract
    This research is an attempt to introduce a connection between graph theory and linear transformations of finite dimensional vector spaces over a field F (in our case we will be considering R). Let Rᵐ, Rⁿ be finite vector spaces over R, and let L be the set of all non-trivial linear transformations from Rᵐ to Rⁿ. An equivalence relation ~ is defined on L such that two elements f, k ϵ L are equivalent, f ~ k, if and only if ker (f) = ker (k). Let V be the set of all equivalence classes of ~. We define a new graph, G([t] : Rᵐ → Rⁿ), to be the undirected graph with vertex set equal to V , such that two vertices, [x] ; [y] ϵ G([t] : Rᵐ → Rⁿ) are adjacent if and only if ker (x) ∩ ker (y) ≠ 0. The relationship between the connectivity of the graph G([t] : Rᵐ → Rⁿ) and the values of m and n has been investigated. In addition, we determine the values of m and n for a complete and totally disconnected graph, as well as the diameter and girth of the graph if connected.
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    http://hdl.handle.net/11073/16476
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