I'm trying to think through the hierarchy of mathematical objects leading to an orthonormal basis. A space may be filled by vectors and a vector may be thought of as a linear combination of components in a basis. But, if a basis is not orthonormal, is each vector constituting said basis not itself a linear combination of components in an orthonormal basis?
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2$\begingroup$ Read your definitions carefully. What is the definition of a "basis"? What is the definition of an "orthonormal basis"? What distinguishes the two? $\endgroup$– Xander Henderson ♦Jan 8, 2022 at 19:12
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1$\begingroup$ Your question doesn't make sense. $\endgroup$– azif00Jan 8, 2022 at 19:15
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$\begingroup$ A basis is a minimum set of vectors whose linear combination may yield all other vectors in a space. An orthonormal basis possesses this same feature but may only be made up of vectors that are orthogonal to one another and of unit norm. $\endgroup$– VinnyJan 8, 2022 at 19:24
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$\begingroup$ Your definitions are a little hand-wavy, but are essentially correct. So... suppose that you have a two dimensional vector space spanned by two vectors $u$ and $v$. Can you think of a way of building two orthonormal vectors which span the same space? Hint: you might take one of the orthonormal vectors to be $u/\|u\|$. Further hint: Gram-Schmidt. $\endgroup$– Xander Henderson ♦Jan 8, 2022 at 19:28
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$\begingroup$ I think so. Any two non-colinear vectors in a two dimensional space may form a basis for said space. However, these two vectors need not be orthogonal nor of unit norm. But, I may decompose these two vectors into linear combinations of two other vectors that do indeed possess these features. Thus, I will have written my original basis vectors in terms of a new basis. Am I correct in saying that an issue with my original train of thought was nesting the second basis within the first since that fails to recognize the first basis as not constituting a space per se? $\endgroup$– VinnyJan 8, 2022 at 19:43
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Yes, you can make a non-orthonormal basis from an orthonormal basis. Having an orthonormal basis is a stronger condition than just having a basis. You might want to look into the Gram-Schmidt process for some insight on the relation of orthonormal basis and non-orthonormal basis.
