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Math Help - [SOLVED] Vector spaces over division rings.

  1. #1
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    [SOLVED] Vector spaces over division rings.

    Hi:
    I have the following definition: let V be a vector space over
    a division ring D. A mapping a of V into V is called a linear trans-
    formation of V if it has the followiwng two properties:
    (x+y)a = xa+ya for x,y \in V,
    (x \alpha)a = (xa) \alpha for x \in V, \alpha D

    And here I find an odd thing. If a is the mapping multiplication by
    a scalar (that is, by an element of D), then a is not in general a linear trans-
    formation of V according to the definition, because D needs not be
    commutative. Any hint will be welcome.
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  2. #2
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    First, in English, at least, a set with operations of addition and "scalar multiplication" with the scalars from a division ring rather than a field is usually called a "module", not a "vector space".

    Second, If you define scalar multiplication by " x\alpha" for \alpha in D, then you must either define \alpha x= x\alpha or leave [tex]\alpha x[tex] undefined. The fact that multiplication of scalars is not commutative has nothing to do with the relationship between x\alpha and \alpha x.
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  3. #3
    Senior Member roninpro's Avatar
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    Quote Originally Posted by HallsofIvy View Post
    First, in English, at least, a set with operations of addition and "scalar multiplication" with the scalars from a division ring rather than a field is usually called a "module", not a "vector space".
    Actually, Thomas Hungerford defines a vector space using division rings.
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  4. #4
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    Quote Originally Posted by ENRIQUESTEFANINI View Post
    Hi:
    I have the following definition: let V be a vector space over
    a division ring D. A mapping a of V into V is called a linear trans-
    formation of V if it has the followiwng two properties:
    (x+y)a = xa+ya for x,y \in V,
    (x \alpha)a = (xa) \alpha for x \in V, \alpha D

    And here I find an odd thing. If a is the mapping multiplication by
    a scalar (that is, by an element of D), then a is not in general a linear trans-
    formation of V according to the definition, because D needs not be
    commutative. Any hint will be welcome.
    multiplication by an element of D is a linear transformation iff that element is in the center of D.
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  5. #5
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    Quote Originally Posted by roninpro View Post
    Actually, Thomas Hungerford defines a vector space using division rings.
    And Neal H. McCoy too.
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  6. #6
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    Quote Originally Posted by NonCommAlg View Post
    multiplication by an element of D is a linear transformation iff that element is in the center of D.
    Quite understandable. And if D is a field the center of D is D. So in this case that mapping is always a linear transformation. Thanks a lot, NonCommAlg.
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