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Thread: Linear map help

  1. #1
    Super Member
    Sep 2008

    Linear map help

    i need help with part a and b, not sure if they are or are not linear or not,

    for part c) can i set x = 0 and y = , which gives me 2, which does not equal 0, so part c is not a linear map?

    also part d, is not a linear map because if a =0 , b = 0, and c= 0, than we get 0, in the transformation T, so it is a linear map ?

    Thank you.
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  2. #2
    Senior Member jakncoke's Avatar
    May 2010

    Re: Linear map help

    you are thinking too hard. Just try to test if the operator preserves linearity, meaning $\displaystyle T(a+b)=T(a)+T(b)$.
    for a)Trace of a matrix is just adding the diagonal entries. is Trace(A*B)=Trace(A)*Trace(B). For example take $\displaystyle \begin{bmatrix}1 & 0 \\ 0 & 1 \end{bmatrix} * \begin{bmatrix}1 & 0 \\ 0 & 0 \end{bmatrix}$ Trace of first matrix is 2, trace of second matrix is 1 but A*B = $\displaystyle \begin{bmatrix}1 & 0 \\ 0 & 0 \end{bmatrix}$ whose trace is 1. so Trace(A*B)=1 which does not equal Trace(A)*Trace(B)=2, so Tr is not a linear transformation.

    b)$\displaystyle (AB)^{t} = B^{t}A^{t}$ so the linear map given by $\displaystyle A \to A^{t} $, so AB under this map goes to $\displaystyle (AB)^{t} = B^{t}A^{t}$ but this isnt equal to A under the map, times B under this map which is $\displaystyle A^{t}*B^{t} \not = (AB)^{t} = B^{t}A^{t}$ so this map isnt linear.

    u can do the rest.
    Thanks from Tweety
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