Determine if the transformations are linear operators?

a) T(x, y) = (2x, y)


b.) T(x, y) = (-y, x)


c) T(x, y) = (x+1, y)





How would you determine if these are linear operators? I know that these conditions must satisfy: T(u+v) = T(u) + T(v)


T(ku) = kT(u)


But I do not understand how to apply these properties.

Determine if the transformations are linear operators?
Let u=(u1,u2) and v=(v1,v2) and a,b be scalars for each case. Then for each case, it suffices to determine whether T(au+bv)=aT(u) + bT(v):





a) T(au+bv) = T(au1+bv1,au2+bv2)


= (2(au1+bv1),au2+bv2)


= (2au1,au2)+(2bv1,bv2)


= a(2u1,u2) + b(2v1,v2)


= aT(u) + bT(v), so yes, this is linear.


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b) T(au+bv) = T(au1+bv1,au2+bv2)


= (-au2-bv2,au1+bv1)


= (-au2,au1) + (-bv2,bv1)


= a(-u2,u1) + b(-v2,v1)


= aT(u) + bT(v), so yes again.


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c) T(au+bv) = T(au1+bv1,au2+bv2)


= (au1+bv1+1,au2+bv2)


= (au1+1,au2) + (bv1,bv2).


Note that this is NOT the same as aT(u) + bT(v), so T is NOT linear.


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daisy