Question

5. Let B = {{1_21-63 21:13 31-_ )} and A = [5_] a. Prove that B is a basis for M22- b. Considering B as an ordered basis, find (A), the coordinate vector of A relative to B.

Answer 1

B-{[18]. 1:870:1 61 and A-G!] (6) We have to prove that B8 basis fole Mae fost we will show that B is linearly inclepterdent set. G let a,b,c,d ER Such that al 3] + 66 ]+(3 37+2 [ ] = 6:1 > fa+d 20+2b+30 arabiad -2a+bt2d To of a+d=o - o 2a+2b+30=0 - a +36 +3c-d=0 -2a+b+2d -0 - By we have a= d 6 Using this in and 6 we get b=-40 using ② & in we get 20+2(-4d) +3c=0 > 3c = lod ③ Using 6 6 8 in gives -d +3(-4d) lod-d=0 ce udo ► do agb=0 (by , 680) So the given set B is linberly independent. 6

111111111111111129229999 Now, dimension of Moe is 4 and dimension of Bris also 4. So any 2x2 mateix Can be expressed as linear Combination of elements of B. So, B is spanning set for Moxo of May Since is planning set and lineary independent so B. & a basis for Max (6 we have to find Caldinate vector of A relative to Be ie for abcd ER such that a 1 2] +b] + 37:04]=68_ we have to find a, b, c, d, Nao we have Gtd = 1 20+2b+c=2 a +36 +3c-d=3 -2a+b+ 20 = 4 - from 0, we have a=1-d using in 9 al get a -2(-)+b+ 2d = 4 > b=6-ud - Using and 6 in , we get 2(1-2) + 2(6-4d) +36=2 & 3c = -12 + lod - 1 using , in ., we get 1-2 +3 (6-4d) + (-12+10d)-d=3 → d1 > G-0,b=2, c= -22, d = 1