Question

Find the orthogonal projection of v⃗

26 11 8 4 0 (1 point) Find the orthogonal projection ofv- 0 onto the subspace V of R spanned by and 28 (Note that these three vectors form an orthogonal set.) projv (u)-

Answer 1

v^rightarrow = [26 1 - 22 - 28], [1 0 4 - 8] [8 0 - 4 - 1] [4 0 7 4] orthogonal projection rightarrow proj_v (v^rightarrow) = v - v_1/v_1 v_1 v_1 + v.v_2 v_2 v_2 v_2 + v.v_3/v_3 v_3 v_3 v^rightarrow = [26 1 - 22 - 28] v_1 = [1 0 4 - 8] v_2 = [8 0 - 4 - 1] v_3 = [4 07 4] v.v_1 = [26 1 - 22 - 28] [1 0 4 - 8] = (260) (1) + 1(0) + (- 22) (4) + (- 28) (- 8) = 162 v_1.v_1 = [1 0 4 - 8] [1 0 4 - 8] = 1 + 0 + 16 + 64 = 81 v.v_2 = [26 1 - 22 - 28].[8 0 - 4 - 1] = 26(8) + 1(0) + (- 22) (- 4) + (- 28) (- 1) = 324 v_2.v_2 = [8 0 - 4 - 1] .[8 0 - 4 - 1] = 64 + 0 + 16 + 1 = 81 v.v_3 = [26 1 - 22 - 28] [4 0 7 4] = (26) (4) + 0 +(- 22) (7) + (- 28) (4) = -162 v_3.v_3 = [4 0 7 4] [4 0 7 4] = 16 + 0 + 49 + 16 = 81