Complete the proof of Theorem 514 Orthogonal projections are

Complete the proof of Theorem 5.1.4: Orthogonal projections are linear transformations.

Solution

Let W be an inner product space and V be a subspace such that V V = W . Then we can define the operator PV of orthogonal projection onto V. Namely, any vector x W is uniquely represented as x = p + o, where p V and o V , and we let PV (x) = p.

Take any vectors x, y W . We have x = p1 + o1 and y = p2 + o2, where p1, p2 V and o1, o2 V . Then x + y = (p1 + p2) + (o1 + o2). Since p1 + p2 V and o1 + o2 V , it follows that PV (x + y) = p1 + p2 = PV (x) + PV (y). Further, for any scalar r we have rx = rp1 + ro1. Since rp1 V and ro1 V , we obtain PV (rx) = rp1 = rPV (x). Thus PV is a linear operator.