(2) Let V(vector) = (4, 3, 2, 1) 2 R4. For each basis B of R4 given below, find
ID: 3099748 • Letter: #
Question
(2) Let V(vector) = (4, 3, 2, 1) 2 R4. For each basis B of R4 given below, find the corresponding
vector [V(vector) ]B 2 R4.
(a) B = {(1, 0, 0, 0), (1, 1, 0, 0), (1, 1, 1, 0), (1, 1, 1, 1)}
(b) B = {(1, 0, 2, 0), (0, 3, 0,-1), (1, 1, 1, 1), (1, 0, 0, 1)}
Explanation / Answer
So V = (4, 3, 2, 1) in R4. (a) The new basis is vectors w,x,y,z: B = {W=(1, 0, 0, 0), X=(1, 1, 0, 0), Y=(1, 1, 1, 0), Z(1, 1, 1, 1)} V here is obvious but I will show the normal method to calculate: V = (4,3,2,1) becomes V = aW + bX + cY + dZ implies: a.w1+b.x1+c.y1+d.z1 = v1 = 4 a + b + c + d = 4 a.w2+b.x2+c.y2+d.z2 = v2 = 3 b + c + d = 3 (because w2=0) a.w3+b.x3+c.y3+d.z3 = v3 = 2 c + d = 2 (because w3,x3=0) a.w4+b.x4+c.y4+d.z4 = v4 = 1 d = 1 (because w4,x4,y4=0) Now d=1 and c+d = 2 => c=1. Now c=1 and d=1 and b+c+d = 3 => b=1. Now b=1 and c=1 and d=1 and a+b+c+d = 4 => a=1. So V in new base is V=(1,1,1,1) (b) B = {W=(1, 0, 2, 0), X=(0, 3, 0,-1), Y=(1, 1, 1, 1), Z=(1, 0, 0, 1)} Same technique: V = (4,3,2,1) becomes V = aW + bX + cY + dZ implies: a.w1+b.x1+c.y1+d.z1 = v1 = 4 a + c + d = 4 (because x1=0) (eq.1) a.w2+b.x2+c.y2+d.z2 = v2 = 3 3b + c = 3 (because w2=0, x2=3, z2=0) (eq.2) a.w3+b.x3+c.y3+d.z3 = v3 = 2 2a + c = 2 (because w3=2,x3=0,z3=0) (eq.3) a.w4+b.x4+c.y4+d.z4 = v4 = 1 -b + c + d = 1 (because w4=0,x4=-1) (eq.4) Now the 1st and 4th equations, doing (1st)-(4th) give: eq (5) a+b =3 OR a = 3-b Carrying this (5) into the 3rd equation (2a+c = 2) gives: equation(6) 6-2b+c =2 or c=2b-4 And taking this (c=2b-4) to equation 2 (3b+c = 3) we get: 3b+2b-4 = 3 OR 5b = 7 Now b = 7/5 => c = -6/5 Now c = -6/5 => a = 8/5 And finally going back to equation (1) we get: 8/5 – 6/5 + d = 4 OR d = 18/5 So V in new base is V=(8/5 , 7/5, -6/5, 18/5) Verify: B = {W=(1, 0, 2, 0), X=(0, 3, 0,-1), Y=(1, 1, 1, 1), Z=(1, 0, 0, 1)} (8/5)W + (7/5)X – (6/5)Y + (18/5)Z = (8/5,0,16/5,0) + (0,21/5,0,-7/5) - (6/5,6/5,6/5,6/5) + (18/5,0,0,18/5) So this is = ( [(8-6+18)/5], [(21-6)/5] [(16-6)/5], [(-7-6+18)/5] ) = (20/5, 15/5, 10/5, 5/5) = (4,3,2,1)Related Questions
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