Question

Find a basis B for the domain of T such that the matrix of T(x, y) = (3x + 3y, 3x + 3y) relative to B is diagonal. a B = {(1, -1), (1, 1)} b B = {(1,0), (0, 1); c. B = {(1, 0), (1, 1); d. B = {(1, -1), (1, 0)) e B = {(0, 1), (1, 1);

Answer 1

Find basis B а such that the for the domain of T matríme of Tcx,y) = (3x+3y, 34+34) is diagonal. relative to B Solution: Given T: R² R is defined by T(x, y) = (3x+3y, 3x+3y) standard know the that basis of R is {e, es} Here e = [] e 2 Now T(1, 0) =(3, 3) = 3.6, +3.62 TCO, D =(3, 3)=3. e, +3-12 The standard matrix of T = 3 =A 3 (say The characteristic polynomial of A be written can as

தினம் டப்பிறப்பு பிறப்பு மந்தநாள் 3 det CA-XI) 3-x இனம் 3 3-x = (3-x3²9 =9-6x+x2-9 திரப் - 2 x² - 6x umpa = x (x-6) பமும் 1A- xIl=0, we get Letting 16 x=0, x=6. The eigen values of A are roots of JA-xIl=0. the To The eigen values of A=0,6. find the eigen vectors:- * solving the system (A-OT)x=0 get eigen vector corresponding we to X = 0.

3310 The augmented matrix is 13 3 Now, 3 O 3 O 3 3 R₂ R3 R *** olo 3 3 3x, +3x2 = 0. 1:22 - The eigen vector 5

* Solving we the rector the equation CA-6I) x = 0, get Corresponding eigen for x=6. augmented matrix 3-6 The 3 3 3-6 -3 3 O 3 -310 Now -3 Rg->R2 + R} 3 -3 O olo Now the system transforms into -32, +3x2 = 0 corresponding eigen vector for x=6 95 The The basis B for the domain - CP. q) Ans :a:{(1, 1), (1,1)}] 67