Inner Product spaces - Orthonormal projection

(d) Inner product spaces - Orthonormal projection

Problem 1.

In each of the following parts, find the orthogonal projection of the given vector on the given subspace W of the inner product space V.

Find the distance from the given vector to the subspace W.

Solution :

Let V be an inner product space and W be the subspace of V.

(a) Let the vector space V = R², vector u = (2, 6) and subspace W = {(x, y): y = 4x}.

Find the orthogonal projection of the vector u on the subspace W as follows.

We know that, the orthogonal projection of u is <u, v₁> v₁

Thus, the required orthogonal projection of u is 

Suppose, v is the orthogonal projection of u, then the required distance of the given vector u to the subspace W is the length of u - v.

(b) Let the vector space V = R³, vector u = (2, 1, 3) and subspace,

W = {x, y, z : x + 3y - 2z = 0}}

Find the orthogonal projection of the vector u on the subspace W

We know that the orthogonal projection of u is

Now, the required orthogonal projection of u is as follows:

Thus, the orthogonal projection of u is 

Suppose v is the orthogonal projection of u, then the required distance of the given vector u to the subspace W is the length of u - v.

(c) Let vector space V = P(R), with the inner product.

Now, the required orthogonal projection of h is as follows:

Thus, the orthogonal projection of u is 

Suppose V is the orthogonal projection of g, then the required distance of the given vector g to the subspace W is the length of g - v

Hence,