Question

Q3. Determine whether the set of vectors in P2 is linearly dependent or linearly independent. S= {2 - x, 4x – x², 6-7x + x>). Q4. Show that the following set is a basis of R. --00:07)}

Answer 1

Soluhon: 3) The giren set ils - S = JQ-X, 40 -x+; 6-4x+x?}. We know that B-0114.25 is a standard basis for P2. Now, determu'nant of the co-effecients of 1.x, of rectors in s is given by - 12 0 6 = 2(4-7) + 0(0+1)+ 6(1-0) - l = 2(3)+0+ 6 = -6 +6 so, the vectory in Š are linearly dependent. 4) To show that the set sa $(12,3), (012),1-2,0.1) is a basi No. of vectors in S = 3 = dimension of R3. Also, the determinant of beekee the matrix formed by the rectors of S is 1 2 3 O 2 -20 1 = 1(1-0)+20-4-0) + 3 (0+2) = 1-8+6 3-1 + 0 So, the rectors of s are linearly indepenelent. Hence, s is a basis of R² X --