Does every vector space have an orthonormal basis

1. Does every vector space has an orthonormal basis?
2. Do all subspaces have an orthonormal basis?
3. How many orthonormal basis can a vector space have?
4. What is an orthonormal basis of a vector space?

Does every vector space has an orthonormal basis?

Every finite-dimensional inner product space has an orthonormal basis, which may be obtained from an arbitrary basis using the Gram–Schmidt process. can be written as an infinite linear combination of the vectors in the basis.

Do all subspaces have an orthonormal basis?

Is it true that every subspace of Rⁿ has an orthogonal basis? The answer is yes, and the reason is that, starting with any basis, we can construct an orthogonal one via the following algorithm.

How many orthonormal basis can a vector space have?

If you're asking how many different orthogonal sets of vectors can serve as a basis, then even in 3D space the answer is infinitely many.

What is an orthonormal basis of a vector space?

A basis is orthonormal if all of its vectors have a norm (or length) of 1 and are pairwise orthogonal. One of the main applications of the Gram–Schmidt process is the conversion of bases of inner product spaces to orthonormal bases.