Thanks for helping Attempt Results Entered Answer Preview 40

Thanks for helping

Attempt Results Entered Answer Preview 4.086 4.086 0.848 0.848 3.796 3.796 3.897 3.897 he answer above is NOT correct find the closest point to Vin the subspace W spanned by and (1 point) ven V 10 4.086 0.848 3.796 3.897 12 Result incorrect

Solution

Because v1 · v2 = 0 +28 -16 -12 = 0, {v1, v2} is an orthogonal set. Beause neither v1 nor v2 is 0, this means that {v1, v2} is linearly independent. Since W is spanned by {v1, v2}, we have that {v1, v2} is an orthogonal basis for W. Therefore, the answer is ProjW x, and we can calculate it as:

{(x.v1)/(v1.v1)}v1 + {(x.v2)/(v2.v2)}v2

{(-24 +16 -40 +1)/(36 +16+16 +1) }v1 +( 0 +28 +40 -12)/( 49 +16 + 144)v2

= -0.6811v1 + 0.268v2

= ( 4.08 , 2.74 , -2.74 ,0.68) + (0 , -1.87 ,-1.072 , -3.21)

= (4.08 , 0.87 , -3.812 , -2.53)