Question

Q8 6 Points Let T : R2 + Rº be a linear transformation with PT(x) = x2 – 1.

Answer 1

as x ²1 = Giren T: 18² 18² with pr(a) = x² 1. We have to determine whether T is always diagonalizable Ans As pq (x) = -1 :: we have chanacteristic polynomial (x-1) (x+1) Also dim t = 2. we have a sussult that if charaderistic polynonual of a linear transformation has dinear factors i.e. factors of degree 1 fou each eigenvalue then I is always diagonalizable 1-1 and both are linear is always diagonabirable. 4-4 7:1R² IR? vs T(T(U) In this question we can take any linean transformation on IR? Example 1 T(ay) = (-x, y) TR² R² Then (0.4)T(T(X,Y)) = = T(-x, y) = (-(-x), Y) AS ot! P - kry

Then T²X) = T(AX) = A(AX) = Ax. In this 6 (y (xy . Te is identity transformation, Example 2 T: 1R² R² T (24) paukt by] =(a ] Lcxtdy- Tex,y) = (2x+3y x+2y Then T2: 1R² R² I say (A) T2 A²X AS TX) = AX :. They AC :14] -*;] 7 ; 7 J