Homework SourseShow that B v1 v2 v3 ia a basis for Ropf3 where v1 4 1 0 v
Show that B = {v_1, v_2, v_3} ia a basis for Ropf^3, where v_1 = [4 1 0], v_2 = [-7 8 0], v_3 = [1 1 1]. Express v = [-9 -15 -3] as a linear combination of the basis vectors v_1, v_2, v_3.![Show that B = {v_1, v_2, v_3} ia a basis for Ropf^3, where v_1 = [4 1 0], v_2 = [-7 8 0], v_3 = [1 1 1]. Express v = [-9 -15 -3] as a linear combination of the](/WebImages/39/show-that-b-v1-v2-v3-ia-a-basis-for-ropf3-where-v1-4-1-0-v-1119089-1761595094-0.webp "Show that B = {v_1, v_2, v_3} ia a basis for Ropf^3, where v_1 = [4 1 0], v_2 = [-7 8 0], v_3 = [1 1 1]. Express v = [-9 -15 -3] as a linear combination of the")
Solution
vector v can be expressed as : v = xv1 + yv2 + zv3
Now, (-9, -15 , -3) = x( 4, 1, 0) + y( -7, 8 , 0) +z ( 1, 1, 1)
-9 = 4x -7y +z
-15 = x + 8y +z
-3 = z
Now solve the three equations to get x, y , z:
x = -74/13 ; y = -20/13 ; z= 3
So, v = -74v1/13 - 20v2/13 + 3v3
