Orthogonality question from Linear Algebra

Question

a) Let u1,u2,u3 are vectors in RN. Suppose that vectors u1, u2 are orthogonal and the norm of u2 is 4 and uT2.u3 = 7. Find the value of the real number 'a' in u1= u2 + au3.

b) Let S = {v1,v2......vk} be a set of nonzero vectors in RN. Suppose that S is an orthogonal set. Show that S is linearly independent.

Answer 1

a) It is given that u₁ is orthogonal to u₂, so u₁ • u₂ = 0, that is, (u₂ + a u₃) • u₂ = 0. Distributing the dot product over addition, we find u₂ • u₂ + a u₃ • u₂ = 0. Since the norm of u₂ is 4, u₂ • u₂ = ||u₂||² = 4² = 16. Thus 16 + a(7) = 0, or, a = -16/7.

b) Let c₁,...,c_k be real numbers such that c₁v₁ + ••• + c_kv_k = 0. Let j be an integer between 1 and k. Orthgonality of S implies v_j • v_i = 0 if i ≠ j. Taking the dot product of the linear dependence equation with v_j, we obtain c_jv_j • v_j = 0, or c_j||v_j||² = 0. Since v_j is nonzero, then c_j = 0. As j was arbitrary, this shows that S is linearly independent.