plz help me answer it with steps

i am also attaching the answer key

ANSWERS: 2. a) Hint: Use the fact that the determinant of a matrix with integer entries is an integer. bj () k -0 (0)k = Dandk =6 (RIjk =6. 312] 3. a) 021 001 4. 2)- bj (1,1,-2) = 5.a) p = 2 b) x = 3 - 3s, y = $, 2 = [ are the parametric equations and z + 3y - 3 = 0 is the general equation. c) C = (0, 112v1946 5 ", 0) 6. 3) 2 5 bj - ()-8 7. alz bj Oxy = 12 = t where tell 1121 #. Hint: It is enough to show that -1 0 1 = 0 211 9. a] Hint: It is enough to show the linear independence. bj (1,2,-1) c) (x', x-x',3] 10. a) False b) True () False8. Let W = S'p{(1,1,2),(H1,0,1),{2, 1,1,)}. :1) Show that{[1,1,2),(1,U,1),(2, 1,1)} is linearly dependent in IRE. b) Find a basis and the dimension of W. {2) Find a system ofhomogeneous linear equations whose solution space is W 9. a) Show that [[3 i] , [i 1], [:31 31] } is a basis of the vector space ofal] 2X2 symmetric matrices. 3 2 C) Find a basis of P2 in which the vector x + 3 has all coordinates equal to 1. b) Find the coordinates of the vector [3 ] relative to the basis given in part (a)