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Math Help - Linear Transformations

  1. #1
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    Linear Transformations

    I have to either show that a transformation is either linear or not due to the additivity or the homogeneity.

    The first question is:
    T(x,y)=(x+1,y)

    I answered:

    T(x)+T(y)= mx+my
    m(x+1)+my= mx+m +my
    mx+my =/= mx+m+my
    therefore
    T(x,y)=(x+1,y) is not linear under additivity.

    How would I prove or disprove that the transformation is linear under homogeneity?
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  2. #2
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    Your proof does not make sense because T(x) is not defined.

    However if \vec a=(a_1,a_2) \mbox{ and }\vec b=(b_1,b_2)
    So
    T(\vec a + \vec b)=T(a_1+b_1,a_2+b_2)=(a_1+b_1+1,a_2+b_2)
    Whilst
    T(\vec a) + T(\vec b)=(a_1+b_1+2,a_2+b_2)

    For the second part.
    T(c\vec a)=T(ca_1,ca_2)=(ca_1+1,ca_2)
    but
    cT(\vec a)=(ca_1+c,ca_2)
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  3. #3
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    Quote Originally Posted by whipflip15 View Post
    Your proof does not make sense because T(x) is not defined.

    However if \vec a=(a_1,a_2) \mbox{ and }\vec b=(b_1,b_2)
    So
    T(\vec a + \vec b)=T(a_1+b_1,a_2+b_2)=(a_1+b_1+1,a_2+b_2)
    Whilst
    T(\vec a) + T(\vec b)=(a_1+b_1+2,a_2+b_2)

    For the second part.
    T(c\vec a)=T(ca_1,ca_2)=(ca_1+1,ca_2)
    but
    cT(\vec a)=(ca_1+c,ca_2)
    So I'm guessing I sorta combined the two into an epic fail? I think I understand... Probably not.

    A second question is:
    T(x,y)=(y,y)

    For this, I would have:
    let \vec a=(a_1,a_2) \mbox{ and }\vec b=(b_1,b_2)
    then
    T(\vec a + \vec b)=T(a_1+b_1,a_2+b_2)
    and
    T(\vec a) + T(\vec b)=?
    I'm not sure what needs to be done here...

    For the second part, I'm not sure how to do what you did above using two different vectors, because I would be comparing x to y at some point, correct?
    Sorry for my idiocy, by the way.
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