Find linear transformations U, T:F2→F2 such that UT=To (the zero transformation) but TU≠T0.Use your answer to find matrices AandB such that AB=0 but BA≠0.

Question

Find linear transformations U, T:F2→F2 such that UT=To (the zero transformation) but TU≠T0.Use your answer to find matrices AandB such that AB=0 but BA≠0.

jlo2niT · Accepted Answer

Step 1 Here the main objective is to find the linear transformation U,T:F2F2 such that UT+T0 but TU is not equal to T0 Where, T0 is the zero linear transformation Step 2 Here on defining as follows, T:F2→F2 by T(x+y,0) for all (x,y)∈F2−(1) And U:F2→F2 by U(x,y)=(y,y) for all (x,y)∈F2−(2) On computing UT UT(x,y)=U(T(x,y)) By using (1) UT(x,y)=U(x+y,0) By using (2) UT(x,y)=T0 Therefore UT=T0forall(x,y)∈F2 Step 3 On computing TU TU(x,y)=T(U(x,y)) By using (2) TU(x,y)=T(y,y) By using (1) TU(x,y)=(y+y,y)=(2y,y) Here,TU(x,y) is not equal to T0Therefore,TU is not equal to T0 for all (x,y)∈F2 Step 4 On calculating the matrices of the linear transformations U and T The standard basis for R2 is {(1,0),(0,1)} First on computing the matrix of U Thus the definition of U is U(x,y)=(y,y) U(1,0)=(0,0) =0(1,0)+0(1,0) U(0,1)=(1,1) =1(1,0)+1(0,1) Therefore the matrix of U with respect to the standard basis is A=[1100] Step 5 On calculating the product of AN and BA as follows, AB=[0101][1100] =[0(1)+(1)00(1)+1(0)0(1)+1(0)0(1)+(1)0] [0+00+00+00+0] [0000] =0 Step 6 The product of BA is as follows, BA=[1100][0101] AB=[0101][1100] [0+00+01+10+0] =[0200] Therefore BA is not equal to 0 Therefore the matrices A and B of the linear transformations U and T respectively This satisfies AB=0 but BA is not equal to 0

Find linear transformations U, T : F^{2} \rightarrow F2 such that UT =T

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