Question

n Exercises 15–16, find the eigenvalues and a basis for each eigenspace of the linear operator defined by the stated formula. [Suggestion: Work with the standard matrix for the operator.]

16. T(x,y,z)=(2x−y−z,x−z,−x+y+2z)

In Exercises 15-16, find the eigenvalues and a basis for each eigenspace of the linear operator defined by the stated formula Suggestion: Work with the standard matrix for the operator)

Answer 1

sol. The standard matrin for the T12194,2)=(221-4-2,2-2, -3+ 4+2Z) transformation y A = 9 -1 -1] For eigen value, IA- Il=0 2-1 -1 -11 -14 -0 = (2-4) (--(2-0)+1)+1(2-4-1) -1 (1-1)=0"

- 。 | | o | ⇒ (3-) - A + ²41) 4 (1~d) - 1 + (2-) (-1)² - 。 - 1,1,2] Eign valuey ace I, 1, 3. 1 For eigen vector, [A-dI] x=0 A4 A-1, T -1 - 15, 1 o) |, 1 1 113 > 거 기 - 2 + 1 , , • 3 - o 21] | 2 | l 3 : 거2|| 4 3Ts | | 12 | 3 J

if we take Hz and Hg = 1 this eigen space is . then a basis for {[:]:012 "' J6707 At da Looo LO- li -2 R2 H Rilo , , Ti 19.7 R2 + Rzteile LO - -

23 - : د د ه = لا - لا - د 5 : 23 - 22 - 2 and ه- ;- (-) و - ، لا ۔ : . وہ وہ ) لا - * | =x د -Basis for this eigen space is so