On Wed, Jul 30, 2025 at 10:01:00AM +1000, Bruce Kellett wrote: > > Any set of linearly independent vectors forms a possible basis of a vector > space if the number of linearly independent vectors equals the dimension of > the > space (The linearly independent vectors need not form a mutually orthogonal > set, as long as they are linearly independent.) >
True for finite spaces, not necessarily true of infinite spaces. Consider the space of polynomials, then the linear span of xⁿwhen n is even also has dimension ℵ₀, but doesn't span the whole space. It becomes even trickier with unbounded linear operators (such as d/dx used for momentum), of course. The answer is that for physical situations, only well-behaved operators are used, or sometimes rather well-studied operators (like d/dx) in place of their physical bounded equivalents as a computational convenience, whilst ignoring the incovenient nuances. -- ---------------------------------------------------------------------------- Dr Russell Standish Phone 0425 253119 (mobile) Principal, High Performance Coders [email protected] http://www.hpcoders.com.au ---------------------------------------------------------------------------- -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion visit https://groups.google.com/d/msgid/everything-list/aIll2wx5In_ClCBM%40zen.
