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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.

Re: Linear map help
you are thinking too hard. Just try to test if the operator preserves linearity, meaning .
for a)Trace of a matrix is just adding the diagonal entries. is Trace(A*B)=Trace(A)*Trace(B). For example take Trace of first matrix is 2, trace of second matrix is 1 but A*B = 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) so the linear map given by , so AB under this map goes to but this isnt equal to A under the map, times B under this map which is so this map isnt linear.
u can do the rest.