Let T be a linear operator on the vector space R2x2 of 2 x 2 matrices...

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Let T be a linear operator on the vector space R2x2 of 2 x 2 matrices defined by T(X) 1 2 X, for XE R²×2 !! [4 3 Warning: the linear operator T is lemph{not} the one defined by the matrix acting on V = R in the usual way -- it is actually an operator on the 4-dimensional space of 2 x 2 matrices. 1. Find the minimal polynomial of T. Hint: (Iv,T,T2) is linearly dependent. 2. Use the above result to find a basis B = (e1, e2, e3, e4) for V consisting of eigenvectors of T. 3. Using the standard basis B0 = ( ) for R2x2, write the matrix A = M(T, Bo, Bo) of the linear operator T. 4. What is the change of basis matrix P such that РАР-1 is diagonal? Let T be a linear operator on the vector space R2x2 of 2 x 2 matrices defined by T(X) 1 2 X, for XE R²×2 !! [4 3 Warning: the linear operator T is lemph{not} the one defined by the matrix acting on V = R in the usual way -- it is actually an operator on the 4-dimensional space of 2 x 2 matrices. 1. Find the minimal polynomial of T. Hint: (Iv,T,T2) is linearly dependent. 2. Use the above result to find a basis B = (e1, e2, e3, e4) for V consisting of eigenvectors of T. 3. Using the standard basis B0 = ( ) for R2x2, write the matrix A = M(T, Bo, Bo) of the linear operator T. 4. What is the change of basis matrix P such that РАР-1 is diagonal?