Question

In Exercise, find the eigenvalues of each linear operator and determine a basis for each eigenspace. T -6x1 - 5x2 + 5x37 - 12 L-10x1 - 10x2 + 9x3]

Answer 1

- 6x1 - 582 +5x37 -X2 L-1024 - 10X2 +9x3 1-6 -5 hoor! -10 -10 1 - 6 -5 -10 -10 9 het, x be the eigenvalue of the matraix T. det (T-XI) =0 1-6 To 1-10 -5 51 . 1-7 -10 g-> o 7 (6+1) (1+)(-1) +5.0 +5(0-10-101) = 0 = (1-4)(1+1)=0 > 1 = 4,-1,-1 Hence, the eigenvalues of the linear operators T are 14,-1,-17 Now I Let, e be be the eigenvector corresponding to the eigenvalue x1=4 :- TV = 1, ve → (T-AI)v=0

[-10 -5 557 *123381-(a,0-4) L-10 -10 5][x3] 3 L → -1074 -5x2 +5X3 = 0 -5X2=0, -10%4 -10X, +5X3 = 0 >> X2=0, x3 = 234 30- so, .... (toking x=1) 2 corresponding to the Now, het u 62 the eigen vector eigenvalue +2=-1, ::. (T-121 )u= 0 2218- = -5%, -542 +593 = 0, -10 y, -104, +10yz = 0 → Y₂ = Yity, 8-61-64-[:

..4=10l and ] U₂ = be two eigen vector * * * na to algunos T corresponding to the eigen value -1. the eigen space of the Hence, the basis operators T for is 2008]