Projections onto subspaces

In Euclidean 3-space, we projected vectors onto a line through the origin by first decomposing the vector into the sum of a vector parallel to the line and a vector perpendicular to the line. In general inner product spaces, we can project onto subspaces of any dimension.

Let W be a subspace of an inner product space V, and decompose any vector v in V into v = w + w⊥, where w is in W and w⊥ is in W⊥. Then w is called the orthogonal projection of v onto W, and is denoted by projW(v). The vector w⊥ is called the component of v orthogonal to W.

To find the orthogonal projection of a vector onto a subspace, we need an orthogonal basis of that subspace.

If W is a subspace of the inner product space V, and W has an orthogonal basis {w1, w2, ..., wk}, then the projection of any vector v in V onto W is the sum of its projections on w1, w2, ..., wk.

Proof. We have that v = w + w⊥ for some w in W and some w⊥ in W⊥, and we want to find w, the projection of v onto W.

Since W has orthogonal basis {w1, w2, ..., wk}, w is the sum of its projections on those basis vectors. But for any wi, [w, wi] = [v – w⊥, wi] = [v, wi], so

projwi(w) = ([w, wi]/|wi|2)wi = ([v, wi]/|wi|2)wi = projwi(v)

i.e. the projection of w on wi is the same as the projection of v on wi. So v is the sum of its projections on w1, w2, ..., wk.

To project a vector onto a subspace

Find an orthogonal basis of the subspace (use the Gram-Schmidt process if necessary).

Find the sum of the projections of the vector onto each vector in the orthogonal basis.