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    linear transformation

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    تاريخ التسجيل : 05/04/2008

    m14 linear transformation

    مُساهمة من طرف teacher في الخميس 24 يوليو 2008, 3:06 pm


    A linear transformation between two vector spaces and is a map such that the following hold:
    1. for any vectors and in , and
    2. for any scalar .
    A linear transformation may or may not be injective or surjective. When and have the same dimension, it is possible for to be invertible, meaning there exists a such that . It is always the case that . Also, a linear transformation always maps lines to lines (or to zero).


    The main example of a linear transformation is given by matrix multiplication. Given an matrix , define , where is written as a column vector (with coordinates). For example, consider

    (1)
    then is a linear transformation from to , defined by,

    (2)

    Another example is , and the homotopy from the identity transformation to is illustrated above.
    When and are finite dimensional, a general linear transformation can be written as a matrix multiplication only after specifying a vector space basis for and . When and have an inner product, and their vector space bases, and , are orthonormal, it is easy to write the corresponding matrix . In particular, . Note that when using the standard basis for and , the th column corresponds to the image of the th standard basis vector.
    When and are infinite dimensional, then it is possible for a linear transformation to not be continuous. For example, let be the space of polynomials in one variable, and be the derivative. Then , which is not continuous because while does not converge.
    Linear two-dimensional transformations have a simple classification. Consider the two-dimensional linear transformation

    (3)
    (4)
    Now rescale by defining and . Then the above equations become

    (5)
    where and , , , and are defined in terms of the old constants. Solving for gives

    (6)
    so the transformation is one-to-one. To find the fixed points of the transformation, set to obtain

    (7)
    This gives two fixed points, which may be distinct or coincident. The fixed points are classified as follows.

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    m14 رد: linear transformation

    مُساهمة من طرف teacher في الخميس 24 يوليو 2008, 3:07 pm



    [tr][td]variables
    [/td]
    [td] type
    [/td][/tr]
    [tr][td]
    [/td]
    [td] hyperbolic fixed point
    [/td][/tr]


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      الوقت/التاريخ الآن هو الخميس 08 ديسمبر 2016, 1:54 pm