Question

Given the matrix A = 
1 0 −1 1 3

2 6 −1 0 7

−1 6 2 −3 −2

b) If W = span{[1,0,−1,1,3], [2,6,−1,0,7], [−1,6,2,−3,−2]}, find a basis for the orthogonal complement W⊥ of W.

c) Construct an orthogonal basis for col(A) containing vector [1 2 −1]
 .
d) Find the projection of the vector v =[−3 3 1]
onto col(A).

Please show all work and steps clearly so I can follow your logic and learn to solve myself. I will rate your answers for you. Thank you kindly!

Answer 1

solution :- me haue , -3 -2 span f [1,0,-1,1,3], [2, 6, -1, 0, 1), [-1,6, 2,-3-2]} Let [x1, _, "s, xy, u5] E Wt srice w+= { v E R: <v, w> = 0 , tor all w e = {vER: wvT = 0, for all wE W}, re X3 + xy+ 3X5 =0 + 7x5 + 672 -X3 2.24 + 6%2 +2y - 3xy -215 = 0 - 3 2. where Ax = 0 => 3. Corader, -2 2. - 1 -3 -2 3 I3 -1 xy -3Xy 24 :. [, , , M, r6] = [ --3,+ , , = ?s [1,{,0,0] +zu [-l, {, O, l,0]+ 2s[-3;t 0,1] KE

[ where W3 X3 W3 + Xy Wy + 15 Ws [1,-t, 1,0, 0] , W; = (-3, -, 0,0, 1] fws, wy, ws) is linearly indefendent aet fotlows from o {ws, ww, Wsy Clearly, that and it generates a basio hor wt. io { W3, We, Ws y [I 2 -1) et vi set Containing all linear combinatiomo Cof (A) io of columno of A. (0, 6, 6] , V3 - the [-1, -t, 2), vy = [ 1,0, - 3), vs -[3,7-2) let V2 = [v, V V3 Vy Vs] that so Col CA). basis for fied to me hane finatly, that dear It is span q Vi, Vzy V3, Vy, Vs}. Col(A) Conader, V3 -2 -3 Vụ -2 V5

- 1 V2 V3 Vs V3, Vy , Vs E span { v, V dinearly wrere { v, v»} is span fv, Vz) Col (A) independent set. bor Cof CA). { v, , v_} io Cowstruct Orthogoual baris for CollA), a basis To V, = (1 2 -1] - (0, 6, 6) - G,2,-1) V - <Vz, Uy) y [0, 6,6] - [I,2,-1] = (-1, 4,7) for Col(A). etho gonaf bersio {4 , 2} is O v= [-3, 3, ) Projection of vonto V = <v, l47 ly dcoecA) is (OCCA) Projans ! Co, 6, 6) 2 [1,2,-1) + 22 (-1, 4,7) = ! Co, 6,6) %3D 66 [0,2,2) Proj v