Let u1 u2 uk be an orthogonal basis for a subspace W of Rn

Let {u_1, u_2, ..., u_k} be an orthogonal basis for a subspace W of R^n. Show that for each y in W, the weights in the linear combination y = c_1 u_1 + c_2 u_2 + + c_k u_k satisfy c_i = (y, u_i)/(u_i, u_i) for i = 1, 2, ..., k. Let U = [u_1, u_2, ..., u_k] be a matrix whose columns are formed by an orthonormal basis for W. Show that U^T U = I, what is the size of the resulting identity matrix I? Show that ||U_x|| = ||x|| for all x R^k. An orthonormal basis is an orthogonal basis where each vector in the basis has length 1.

Solution

a. Since, { u₁, u₂ , ...u_i , ..., u_k } is an orthogonal basis for a subspace W of Rⁿ , therefore, u_i . u_j = 0 for i j. Now, [ < y, u_i > / < u_i , u_i > ]= [ ( y.u_i )/ ( u_i . u_i ) ]= [ ( c₁u₁ + c₂ u₂ +... + c_i u_i + ...+ c_k u_k ).u_i ]/ ( u_i . ui ) = c_i ( u_i . ui )/ ( u_i . ui ) = c_i ( as u_i . u_j = 0 for i j)

b. U = { u₁ , u₂ , ...., u_k } is a matrix whose columns are formed by an orthonormal basis for W. Then II u_i II = 1 for all u_i \'s and also u_i . u_j = 0 for i j. Further, since W is a subspace of Rⁿ , each u_i is a n-vector. Thus U is a n x k matrix so that U^T is a k x n matrix Then U^T U = I is a k x k identity matrix.

II Ux II = II ( u₁x₁ + u₂x₂ + .......+u_k x_k ) II = II x II as each u_i has length 1.