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Math Help - Direct Sum & Linear Mappings

  1. #1
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    Direct Sum & Linear Mappings

    Hi there,

    I was just wondering if anyone could tell me if my answer was somewhat right.

    An example in my textbook asks:

    If T:V X V \rightarrow V is the linear mapping T(x,y)=x-y, determine the kernel and range of T.

    My Answer:
    Let a=(x,y)
    Ker T = {a \in V X V | Ta=0}

    0 =Ta
    =T(x,y)
    = x-y
    y = x

    Therefore, Ker T = {(x,y) \in V X V | x=y}

    Im T = T(V X V) = {Ta | a \in V}
    = T(x,y)
    = x - y \forall x,y \in V

    Make sense?
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  2. #2
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    Oct 2009
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    Quote Originally Posted by manygrams View Post
    Hi there,

    I was just wondering if anyone could tell me if my answer was somewhat right.

    An example in my textbook asks:

    If T:V X V \rightarrow V is the linear mapping T(x,y)=x-y, determine the kernel and range of T.

    My Answer:
    Let a=(x,y)
    Ker T = {a \in V X V | Ta=0}

    0 =Ta
    =T(x,y)
    = x-y
    y = x

    Therefore, Ker T = {(x,y) \in V X V | x=y}

    Im T = T(V X V) = {Ta | a \in V}
    = T(x,y)
    = x - y \forall x,y \in V

    Make sense?

    The first part does make much sense, the second one much less: the image is the whole of V (why?)

    Tonio
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