Let R have the Euclidean inner product, and let W be the subspace spanned by the vectors uj = (1, 0, 1, 0), u2 = (0, -1, 1, 0), and u3 = (0, 0, 1, 1). Use the Gram-Schmidt process to transform the basis {u1, u2, u3} into an orthonormal basis. (-. -N6, NG 0), v3 =(-5 5 5 V6 Vo Vo (A) v1 0), v2 = Vo vo 0), v3 = _V3 V3 (B) vị V2 = N6.0), v3 = (-V3 V3 -V5) v2 = (-\5, V6 NG.0), v3 = (-. -. (O vn = (.0. 2.0). %3D V2 = 3 (D) v1 = 6 6 (E) vị 0). v3 = (V3 V3 V3 _V3) V2 6 3 6 _V3 3 V3 (F) vị = V) = 6. 0). v3 = 3 6 2 0). v3 = (-\3 -V3 3 V3) 61 V2 = 3. 6 N N60), v3 = (H) vị = (-E, 0, . 0). 2 6. 13 2.

Apply the Gram-Schmidt orthonormalization process to transform the given basis for R" into an orthonormal basis. Use the vectors in the order in which they are given. B = {(15, 8), (0, 1)} u2 II

The set of vectors {(1, 1, 0), (1, 0, 1), (0, 0, 1)} is an orthonormal basis for the space R^3 in some scalar product <.,.>. Then, the scalar product <(1, 1, 1), (1, 0, 0)> is equal to: (a) -1. (b) 0. (c) 1. (d) 2. (e) 4. The correct answer is (a). Why?

The coordinates for the vector v = (1,2,4) relative to the standard basis B = {(1,0,0), (0, 1,0), (0,0,1)} is [v] B = (1,2,4). The coordinates for v relative to the nonstandard basis B' = {(1, 0, -1), (2,1,-1), (-2, 1,4)} is [v]B¹ = (9,−1,3). Find the transition matrix P-1 from B to B' and show that this does change the coordinate of v relative to B to the coordinate of v relative to B'.