Can a subspace have more than one orthonormal basis?

Can a subspace have more than one orthonormal basis?

Yes I can. So the answer is yes, I can have more than one basis?

Does every subspace have an orthonormal basis?

An orthogonal set of unit vectors is called an orthonormal basis, and the Gram-Schmidt procedure and the earlier representation theorem yield the following result. Every subspace W of Rn has an orthonormal basis.

Does any nonzero subspace have an orthonormal basis?

Theorem. Every nonzero subspace of Rn has at least one orthogonal basis. (In fact, any nonzero subspace has infinitely many orthogonal bases.) The Gram-Schmidt process is an important algorithm which takes a basis for a subspace W ⊂ Rn as input and produces an orthogonal basis for W as output.

Does every Hilbert space have a countable orthonormal basis?

A Hilbert space H is separable (that is, has a countable dense subset) if and only if it has one countable orthonormal basis if and only if every orthonormal basis for H is countable.

Can a subspace have infinite basis?

If V is an infinite dimensional vector spaces, then it has an infinite basis. Any proper subset of that basis spans a proper subspace whose dimension is the cardinality of the subset.

How do you find an orthonormal basis?

Here is how to find an orthogonal basis T = {v1, v2, , vn} given any basis S.

1. Let the first basis vector be. v1 = u1
2. Let the second basis vector be. u2 . v1 v2 = u2 – v1 v1 . v1 Notice that. v1 . v2 = 0.
3. Let the third basis vector be. u3 . v1 u3 . v2 v3 = u3 – v1 – v2 v1 . v1 v2 . v2
4. Let the fourth basis vector be.

Is orthonormal basis unique?

So not only are orthonormal bases not unique, there are in general infinitely many of them.

What is a complete orthonormal basis?

6.56 Complete Orthonormal Systems A complete orthonormal system in a separable Hilbert space X is a sequence {ei}i=1∞ of elements of X satisfying. ( e i , e j ) x = { 1 if i = j 0 if i ≠ j , (where (.,.) X is the inner product on X), and such that for each x ∈ X we have. (32)

How many bases can a subspace have?

Any subspace admits a basis by this theorem in Section 2.6. A nonzero subspace has infinitely many different bases, but they all contain the same number of vectors.