Orthogonal complements as solutions to homogeneous systems of equations

Let be a subspace. How would we explicitly check if a vector is in the orthogonal complement ? When we have general statements like ” is orthogonal to all vectors from ” it is common to check if we can infer them just from the information on a basis. Let’s choose a basis . Then

Proposition

In other words, it is sufficient to check that a vector is orthogonal to elements of a basis (instead of all vectors in ) in order to establish that .

Proof

The proof of is direct: the condition on the right is part of the definition of .

:

What are we given? The condition that is orthogonal to all elements of a basis of . What do we want to show? We want to show that is orthogonal to all vectors in .

Let’s take an arbitrary vector . Because is a basis, can be rewritten as . Then

This completes the proof.

Now, a vector is in the orthogonal complement if it satisfies the finite set of conditions over all . Each of these conditions, when written in coordinates is a linear equation in the coordinates of . More precisely, let

(using that all and are just vectors in ). Then

The vectors are the solutions to these linear homogeneous equations. (In particular, they form a subspace, which is an alternative way to see that is a subspace.) Moreover, the coefficient vectors are linearly independent, because they form the basis of , which means that the space of solutions is a subspace of dimension . We have just shown that

Proposition

Given a subspace