Question

6. Choose one problem, mark it, and solve it. (10 points) Find the closest point to y in the subspace W spanned by v, and v2. (8 points) Is the set of following vectors an orthogonal set of vectors? Justify your answer.

Answer 1

@ Notice that the dot product V, V₂ = 0 that is V. V = VTV2 = [1 -2 -1 2] .4 11 = 1(-4)+(-2)(1) + (-1)(0) +(2)(3) =-4-2 +0+ 6 = 0 V, and V₂ are orthogonal to each other with a subspace spanned by orthogonal Vectors V, and Vz to y in the subspace w is given y (projection of y on W) Hence the closest point by Projw Projwy = Y.V, vt Y. V₂ V.Vi V.V2 (1) (1) +(1)(-2) +(0)41)+(-1) (2) -2 17 (-2) + (-1) 4+ 2² FE 2 3 + (1)(-4)+() (1)+(0) (0)+(-1)(3) (-4)²+1²+0²+ 37 -2 (-6) - + To 26 O 3 2 81 w/w ofw 130 24 65 11 13 6 16 3 10 -6-9 10 13 -84 65

6 Let V- 2 V2 = V3 = ĐOO O 3 2 2 the set of Vectors {v, V2, V3 b is orthogonal if all the following are true v=o, VT=0, TO Compute Ve vz = [-4 103] 4 2 = (-4)0)+ (1) 0) +(0) (4) + (3)(2) = 6 +0 Since the dot product V2 V3 = k Tvz to, Set of Vectols is not an orthogonal set the given