Answer to Question #298448 in Linear Algebra for Maulidi

Question

Let {u1, u2, ..., un} be an orthogonal basis for a subspace W of R

n

, and let T : R

n → R

n

be defined by T(x) = projW (x). Show that T is a linear transformation.

Answer 1

In this section we develop the Gram-Schmidt process, which uses a basis for a vector space to create an orthogonal basis for the space.

The fundamental idea is that if we have a subspace W with an orthogonal basis and some vector u not in W, the vector perpW u = u − projW u is orthogonal to W.

So let us begin with a subspace W of a vector space V , and suppose that B = {v1, v2, ..., vk} is a basis for W. We will construct from this basis an orthogonal basis C = {w1, w2, ..., wk} for W. • First we let w1 = v1, and we define the subspace W1 = span(v1) = span(w1). • Because {v1, v2, ..., vk} is a basis, v2 is linearly independent of v1.

Thus the vector defined by w2 = perpW1 v2 = v2 − projW1 v2 = v2 − hv2, w1i hw1, w1i w1 is a nonzero vector that is orthogonal to W1 = span(w1). We then define W2 = span(v1, v2) = span(w1, w2).

• Next we find a third vector w3 that is orthogonal to both w2 and w1: w3 = perpW2 v3 = v3 − projW2 v3 = v3 − hv3, w1i hw1, w1i w1 − hv3, w2i hw2, w2i w2

Note what has happened here: We have taken the vector v3 and removed the components of it in the directions of w1 and w2. Now we let W3 = span(v1, v2, v3) = span(w1, w2, w3).