i 9. A= |-i 1 - i 1+ i 8. Use gram-schmidt to find the orthogonal basis for the four fundamental subspace

Let V₁ = ,V₂ = 1 and let W be the subspace of R¹ spanned by V₁ and v2. (a) Convert {V1, V2} into an orhonormal basis of W NOTE: If your answer involves square roots, leave them unevaluated. Basis = {}. (b) Find the projection of b = 1 onto W (c) Find two linearly independent vectors in R¹ perpendicular to W. Vectors = { Note: You can earn partial credit on this problem.

Let x = Answer: -3] N and y= -3 Use the Gram-Schmidt process to find an orthonormal basis for the subspace of R³ spanned by * and y. -5 8 Hint: Find an orthogonal basis first, then normalize the vectors.

0 0 Let {U₁ = [²2₂ 92], U₂ = [_-1₁2 24] ²4-[2²4]} U 3 -2 0 0 Use the Gram-Schmidt process to find an orthogonal basis under the Frobenius inner product. Orthogonal basis: V₁ a = Ex: 5 2 0 a 24 C {v=[ ²₂ 2] = [$ ²] × = [-3.37 2]} V₂ V3 2 -2 -2 b 4 d b= = Ex: 5 = c = Ex: 1.23 be a basis for a subspace of R2x2. d = Ex: 1.23

Find the orthogonal projection of the vector

(1) Let W C R³ be a subspace with basis 3 4 1 Use Gram-Schmidt process to produce an orthogonal basis for W.

a = -4 -3 -1 0 and b || -7 -3 11 -12 Use the Gram-Schmidt procedure to determine an orthonormal basis {₁, ₂} for the subspace of Rª spanned by the vectors a and b.