Question: Let 0 < t < 1 be a fixed real number. Let B E M (R) be the matrix M-P) 1-t B = t

Let 0 < t < 1 be a fixed real number. Let

Let 0 < t < 1 be a fixed real number. Let B E M (R) be the matrix M-P) 1-t B = t t1-t and define the linear map TB: R2 R by TBv = Bu for v E R regarded as a column vector. a) Give a formula for Bn for any n and prove it by induction or otherwise. b) What is the matrix of TB relative to the standard ordered basis (e,e2) where e = What is the matrix of T relative to the ordered basis ((t, 1 1), (-1-t, t))? c) What is the kernel of TB? What is the image of TB? Find bases for these subspaces. d) Does TB have eigenvectors in R2? If so, what are they and the corresponding eigenvalues? e) What, geometrically, does the linear transformation TB do to a vector in R? (1,0) and e = (0, 1)T? Hint: The last question is the motivating one, while all the rest are ways to determine the answer to the last one. Perhaps start by finding the eigenvalues/diagonalising the matrix.

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a To find a formula for Bn we can use the concept of diagonalization Lets diagonalize the matrix B b... View full answer

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