Question

7. (a) Use the Gauss-Jordan (and no other) method to calculate the inverse of the 3x3 matrix 1 P= -1 1 2 3 1 [5 marks] (b) Show that the mapping T : R3 R3 defined by Ty 2x + y + z. - x - y + 2z 3x + 2y - z 01-68 9.s +9) is a linear transformation, and write down its matrix representation A with respect to 17 [0] the standard basis 0 Suppose that a new basis {C1,C2,C3} is defined by ei = V1 - 02 = -V1 + 202 + V3 ez = V1 + V2 + 3v3. Write down the transition matrix between the two bases {V1, V2, V3} and {(1,22,23} and using your results in part (a), or otherwise, calculate the matrix representation of Twith respect to this new basis. Hence, or otherwise, calculate T (e,+e2 +e3) in terms of e1, C2, ez [10 marks] e2 =

Answer 1

2 I 0 1 3 on elementary row operation find ?" we will apply the augmented matrix 1 10 2 1 oso Ruthur o 2 1 3 V R3-R₂ R3 Put R; 2 Pj 3 3-2 1-1 -11 1 1 2. oo 11-11 I ĮR + R2 R, 4 3-2 R, R₂ R 1 он 5 4 . 3 3-2 -1 - 1 ng - 3 3 - 2 - 1 1 o 0 1 0 0 L Therefone -3 8-2 17 3

Let Y= r 42 ER² Then for any e, C ER we have Tanit Car, T{ax + CY) = T e, az & C2 Tz Le x3 + C G 2100, 464) +(6,9 + C .)+(99, +3) -Cernitat) - Long + C₂ (2) + 2 (Crez + (283) 30,0,4cxt)+2100, )-(023+043) . Te.com + :) + (24, +7.+13) , (-2, -22 +223) + C₂ (-7 -12 + 273) e, (3x, +222-23) + C2 (34 + 282-33 221 + a₂ +93 T271 + ₂ + 83 - 21-22 +293 t-t₂ +243 32, +29,- 34+24 - 3 ,,T + GT : GT(X) + C₂T(Y) T{ax + Y) = C, TC X9 + GT(%) Therepo ne Tis linear. 2 1 8. Now TV) - T :2u, - 2 + 383 O to on T(₂) =T , - \ +2 V3 O T() = T 4. + 2 v, - V 2 1 A 2 Therefore the matrix representation A = 2 -3

VA Now e, - - 2 17 o o это -70 R2 = V1 + 2 V ₂ + V3 + 0 0 ez a V, + V₂ + 3 4 Let B 1.6.11 and 2 Torfind the transition malnia from B to B'e we will appley elementary now operation on 1 1 -1 2 0 ] 1 3 matroin. such that the left side become identity have pt 5 4 -3 Now from t@ we 3 3-2 "Therefore the transition the transition matroix is 3 3 3 - 2 1 1 VA Now e, tez tez 2 2 4 Tete,+) ace, + Cl2 + Cl₂ 5 3 ce + Cl₂ + Czlz C-C₂ + C3 -e, +222 + 3 C2 + 303

ci-C₂ + 3 2 4 -C, +2C2 + z 2 5 2 + z 2 3 we get C = -5 C =-3 @ 3=6. Ther egone Tle, tez + ez) = -5e, - 3e +6ez.