Problem 1. (10 points) Consider the space P2 of bivariate polynomials of degree not larger than...

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Problem 1. (10 points) Consider the space P2 of bivariate polynomials of degree not larger than two with real coefficients, that is p(x1, x2) = 0+a1x1+a2x2+a3x+a4x+5x12, ai R, i = 0,..., 5 p(x1, x2) EP : and the map L P2 P given by L: p(x1, x2)x2- Op(x1, x2) - X2 Op(x1, x2) Jx2 p(x1, x2) P 1. Argue that L is a linear map (no need to prove it formally, just explain why.) 2. Find the matrix representation M = mat L in the following basis of P2: = span {1, x1, x2, x, x, xx2} 3. Compute the distinct eigenvalues of M, identify their algebraic and geometric multi- plicities, and compute the Jordan form of M. 4. Prove that the subspace CP given by is invariant with respect to L. := span {1, x2, x} 5. Compute the Jordan form of the representation in the given basis of the restriction Ly VV of L to V. Problem 2. (10 points) Consider the parameterized matrix (+1) 0 A = 0 1- 2 0 0 (+1)2) where ER is a parameter. Study the eigenstructure of A (that is, determine the eigen- values of A, their algebraic and geometric multiplicity, the characteristic and the minimal polynomials, and the Jordan form of A) as a function of ER. Problem 1. (10 points) Consider the space P2 of bivariate polynomials of degree not larger than two with real coefficients, that is p(x1, x2) = 0+a1x1+a2x2+a3x+a4x+5x12, ai R, i = 0,..., 5 p(x1, x2) EP : and the map L P2 P given by L: p(x1, x2)x2- Op(x1, x2) - X2 Op(x1, x2) Jx2 p(x1, x2) P 1. Argue that L is a linear map (no need to prove it formally, just explain why.) 2. Find the matrix representation M = mat L in the following basis of P2: = span {1, x1, x2, x, x, xx2} 3. Compute the distinct eigenvalues of M, identify their algebraic and geometric multi- plicities, and compute the Jordan form of M. 4. Prove that the subspace CP given by is invariant with respect to L. := span {1, x2, x} 5. Compute the Jordan form of the representation in the given basis of the restriction Ly VV of L to V. Problem 2. (10 points) Consider the parameterized matrix (+1) 0 A = 0 1- 2 0 0 (+1)2) where ER is a parameter. Study the eigenstructure of A (that is, determine the eigen- values of A, their algebraic and geometric multiplicity, the characteristic and the minimal polynomials, and the Jordan form of A) as a function of ER.