For some reason my book distinguishes the two names.
If a set is an orthogonal set, doesn't that make it immediately a basis for some subspace $W$ since all the vectors in the orthogonal set are linearly independent anyways? So why do we have two different words for the same thing?
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$\begingroup$When you say orthogonal basis you mean that the set is a basis for the whole given space. Every orthogonal set is a basis for some subset of the space, but not necessarily for the whole space.
$\endgroup$ 10 $\begingroup$The reason for the different terms is the same as the reason for the different terms "linearly independent set" and "basis".
Every linearly independent set is a basis for the subspace it spans. But when working in a larger space "basis" means "maximal linearly independent set" (not just spanning a subspace but spanning the whole thing).
An orthogonal set (without the zero vector) is automatically linearly independent. So we have "orthogonal sets" and then maximal ones are "orthogonal bases".
Note: In the end we're essentially just tacking on the adjective "orthogonal". We don't keep the words "linearly independent" in "orthogonal linearly independent set" because they're redundant.
$\endgroup$ $\begingroup$- These two concepts are totally different.
- For "orthogonal set"$M$, we only have Bessel's inequality.
- But, if $M$ is orthogonal basis, then we get the Parseval's theorem. The key point is the completeness of this set M in your space. For example, in finite dimensional space $\mathcal{R}^3$, $\{i,j\}$ is an orthnormal set, but not an orthonormal basis. A common orthonormal basis is $\{i,j,k\}$.
If a set is an orthogonal set that means that all the distinct pairs of vectors in the set are orthogonal to each other. Since the zero vector is orthogonal to every vector, the zero vector could be included in this orthogonal set. In this case, if the zero vector is included in the set of vectors, the set is not linearly independent (thus this set cannot be a basis for some subspace since the vectors in the set are not linearly independent).
So a set being a basis for some subspace and a set being an orthogonal set are two different things. Now, it is possible for an orthogonal set (that does not include the zero vector) to be a basis for the subspace that it spans (and in this case we call that set of vectors an orthogonal basis).
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