Question

(c) Consider the subspace W R4 given by W = ER4 21 +12 + 24 = 0 and x2 + x3 + 14 = 0 14 Find an orthonormal basis H = {h1, h2, h3, h4} for R4 with the property that hy and h2 are elements of an orthonormal basis for W, where orth onormality is defined with respect to the dot product on R4 x R4

Answer 1

Subspace W CIRI Xo1 334 3gp W = {(3) ** 2,43,4x900 € 134 / x, + X₂ + X q = 0 and X₂ + x3. WEIR9. First find basis of subspace Xz= x, Xa= - X, - X2 . O + X₂ C X ( 32 201 -X-X, xa 1 so basises are W? D) and () 2. (0) 0 an V, V₂2 1:0 +0.1 +100+ (-1).(1) 2170 so they are not orthogonal. Let us find orthogonal basis for W by the Gram – schmidt orthogonalization process. Let w, :=VI , Next W₂ 2 V₂t ari where & is a scales to be determined so that wi. Wzzo As w, and we are onthogonal, 02 W, W2 = V. (Vet av.) V, V,2 2 viv, ta vivi a lt a.3 1.1 +0.0 az-/ +1.1 W, 2 V₂ - 1/72 +(-1).(-1) = 3 0 - 0

Now to avoid Practions in computation, let considen 3 W₂ instead of we. Note that the does not change be the onthogonality Scaling 3 W, 23 Thus the set {W,, 3W, 3 is an orthogonal basis for W. However, the length of these Vectors are not 1 as we see 1w1120 +(012+01+(-1), 13 113W,112 V-+(3)*+6-17*+(2) ✓ 1+9+1+4 Now it suffies to normalize the vectons w,,We to obtain an onthonormal basis. Therefore, the set 15 2 2 is an orthonormal basis fon w #9:31 ( hi ha