Given an inner product bases, I know that using a basis, we can obtain an orthonormal basis by applying the Gram-Schmidt process. My question is, is that orthonormal basis unique? If I was to start using another basis, would I get a different orthonormal basis?
2006-10-27 10:04:39 · 3 answers · asked by The Prince 6 in Science & Mathematics ➔ Mathematics
In fact, if you start with the *same* original basis, but in a different order, the Graham-Schmidt process will usually give a different orthonormal basis. Any two orthonormal bases can be thought of as 'rotations' of one another and any rotation applied to one o.n. basis will yield another o.n. basis.
2006-10-27 13:53:33 · answer #1 · answered by mathematician 7 · 0⤊ 0⤋
To illustrate the answer of James claiming that there are infinitely many orthonormal bases, consider the vector space R^2 (the two-dimensional plane). All Cartesian coordinate systems (with the axes forming a right angle) represent orthonormal bases, and you can get a new Cartesian coordinate system from the canonical one (generated by the unit vectors e1 and e2) by arbitrarily rotating it. And you do get different orthonormal bases if you start Gram-Schmidt using different initial bases.
2006-10-27 20:00:10 · answer #2 · answered by ted 3 · 1⤊ 0⤋
There are infinitely many orthonormal bases for any given vector space. The columns of any n-by-n orthogonal matrix (that is, a matrix Q such that Q^T * Q = I) form an orthonormal basis for R^n.
2006-10-27 17:11:00 · answer #3 · answered by James L 5 · 1⤊ 0⤋