Does linear dependence imply span across $\mathbb {R}^n$? As for your other question, yes linear dependence does not imply spanning, see my three vector example above In terms of the specific vector space $\Bbb {R}$, you can have the set $\ {0\}$, which is linearly dependent, but not spanning
ODEs with parameter and continuous smooth dependence on initial . . . For the purposes of differential geometry I would like to fully process the basics of ODEs, namely the existence uniqueness theorem (EUT) and continuous smooth dependence on initial conditions and parameters (CSD), in maximum generality
definition - Linear independence and dependence of vectors . . . But what is linear independence and dependence? Can you please give me trivial examples where I distinctly see the difference between them Why dependence? What does it depend on? Why do I call (in)dependent? Thanks in tons
On the linear dependence of three coplanar vectors The general consensus seems to be that any three coplanar vectors are linearly dependent Here's one source that says so However, considering three vectors, of which two are collinear and the thir
Linear dependence of a set vs a family of vectors I think you're thinking of linear dependence independence as a property of the individual vectors, but it's a property of the family of vectors If you've got a bowl of M Ms that are all different colors-- say, red, blue, and yellow-- you might call it an "all different colors" bowl of M Ms But if you put another blue M M in the bowl, now you don't have an "all different colors" bowl of M