Let 0 < t < 1 be a fixed real number. Let B E M (R)...

 

Transcribed Image Text:

Let 0 < t < 1 be a fixed real number. Let B E M₂ (R) be the matrix M-P) 1-t² B = t² t√1-t² and define the linear map TB: R²2 R² by TBv = Bu for v E R² 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 (e₁,e2) where e₁ = What is the matrix of TÂ relative to the ordered basis ((t, √1 – 1²), (-√1-t², 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 R²? (1,0) and e₂ = (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. Let 0 < t < 1 be a fixed real number. Let B E M₂ (R) be the matrix M-P) 1-t² B = t² t√1-t² and define the linear map TB: R²2 R² by TBv = Bu for v E R² 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 (e₁,e2) where e₁ = What is the matrix of TÂ relative to the ordered basis ((t, √1 – 1²), (-√1-t², 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 R²? (1,0) and e₂ = (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.

Answer rating: 100% (QA)

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