Rank and nullity of linear transformations and matrices

Let T:RT:R3RR2 be the linear transformation defined by T(x,y,z)=(x,z)T(x, y, z) = (x, z), where x,y,zx, y, zRR. Let AA = {(x,y,zx, y, z) ∈ RR3 : T(x,y,z)T(x, y, z) = (0, 0)} and BB = {T(x,y,z):(x,y,z)T(x, y, z): (x, y, z)RR3}.Then
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Let T:RT:R2RR2 be the linear transformation defined by T(x,y)=(x,0)T(x, y) = (x, 0), where x,yx, yRR. Then B=B={(x,y)(x, y)RR2 : T(x,y)=(x,y)T(x, y)=(x, y)} equals
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Let T:RT:R2RR be the linear transformation defined by T(x,y)=xT(x, y) = x, where x,yx, yRR. Then AA = {(x,y)(x, y)RR2 : T(x,y)T(x, y) = 0} equals
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Let T:RT:R2RR2 be the linear transformation defined by T(x,y)=(x,x+y)T(x, y) = (x, x+y), where x,yx, yRR. Let AA = {(x,y)(x, y)RR2 : T(x,y)=(0,0)T(x, y) = (0, 0)}and BB = {(x,y)(x, y)RR2 : T(x,y)=(1,0)T(x, y) = (1, 0)}. Then
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Let T:RT:R2RR4 be the linear transformation defined by T(x,y)=(x,x+y,0,0)T(x, y) = (x, x+y, 0, 0), where x,yx, yRR. Then order of matrix representation of the linear transformation TT is
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