Question

0 5 The set of vectors {x1, x2} spans a subspace W of R3, where x1 = 19- and X 2 -- 2 (a) Use the Gram-Schmidt process to find an orthogonal basis for W. (b) Then normalize this new basis, so that it is an orthonormal basis. (c) Once you've found an orthonormal basis, demonstrate that it is indeed orthogonal after normalization. (d) For a bonus 2 points, calculate a third vector orthogonal to your basis and normalize it to get a basis for R3.

Answer 1

a) 4,= do :10,4,207 ų N₂ - 4,022 u 4, U 19:13):68-701 Orthogonal basis for w= • ŞU, 4₂} Page 1 0 b) To noumalice Via U, 114,11 vão V₂= Uz rios 114211 4 -8 Orthonoanal basis Sveg ug ? c) Vi.v2= OXS Vios + X Tão 4 Vios rios + 16 16 -0 V20x Vos 120 xros as Nov 2=0, hence dathagonal. 1

d). Het y=149 Yag ug ) be third vecton. then Y. 41=0 and you₂ = 0 youe = 0 442 +243 20 43 =- .-242 You q=0 sy, tuy ,-842 = 0 *sy, +442-18)+ (-242) = 0 sy, tuY2+1642=0 oy,= -442 Page 2 take y2=1 then y = (-4,1,-2). vienos Tiyil 16+1+4 4 Basis for R32 {v,, V2, Vs}