Txyxyyx is not a linear transformation and has no kernel a d

T(x,y)=(x-y,y-x) is not a linear transformation and has no kernel a) determine the nulltiy of T b) determine the range of T c) determine the rank of T d) is T one-to-one? why or why not e) in T onto? why or why not

Solution

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

it has no kernel

a) nullity means the rank of kernel.

since it has no kernel hence nullity=0 [answer]

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

hence the image is (x-y,y-x)=(x-y)*(1,-1)=k(1,-1) where k=x-y

hence the image is the LINEAR SPAN OF THE VECTOR (1,-1).

hence the range is R₂   

c) since the range is the LINEAR SPAN OF THE VECTOR (1,-1). hence there is only one independent vector.

hence rank of T=1 [answer]

d) no. it is not one-one

as its nullity is zero.

we have T(4,4)=(4-4,4-4)=(0,0) and also T(5,5)=(5-5,5-5)=(0,0)

hence not one to one.

e) not onto also. because dimension of range is dim(R₂)=2 but T\'s rank=1

hence not onto.