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If $X_1,X_2,\ldots X_N$ be the generators of infinitesimal transformation described by a $N$-parameter Lie group, an infinitesimal group element is given by $$g(\delta\alpha_1,\delta\alpha_1,\ldots)=1-iX_j\delta\alpha_j,$$ and a finite element is given by $$g(\alpha_1,\alpha_1,\ldots)=e^{-iX_j\alpha_j}$$ where in each of the expressions above the repeated index '$j$' is assumed to be summed over the values $j=1,2,\ldots,N$. How can we prove that the generators $X_i$ are all linearly independent?

Solidification
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    A spanning set for an $N$ dimensional vector space consisting of exactly $N$ elements is also linearly independent, hence it is a basis. – peek-a-boo Aug 09 '23 at 12:21
  • How did know that they span the space? From the first equality? – Solidification Aug 09 '23 at 12:33
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    You said they are generators. If not spanning, what else is it supposed to mean? – peek-a-boo Aug 09 '23 at 12:34
  • By generators, I understand $X_i=i\left.\frac{\partial g}{\partial \alpha_i}\right|_{{\alpha_i}=0}$. My knowledge is based on the book by Howard Georgi. – Solidification Aug 09 '23 at 12:35
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    That’s just describing an element of the tangent space, and is not giving much mathematical content. – peek-a-boo Aug 09 '23 at 12:38
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    Ok so I just saw some stuff on Google preview and he says “if the parametrization is parsimonious (that is - all the parameters are needed to distinguish different group elements) the $X_a$ will be independent”. Well this is obviously a vague statement, and actually incorrect. As written he only requires injectivity, but to get an actual honest to god parametrization of a manifold, one imposes conditions like injective, immersion, and homeomorphism onto image. The immersion part is a rank condition which here implies the associated tangent vectors are independent. – peek-a-boo Aug 09 '23 at 12:46
  • Prof. Howard Georgi has published more than one book. – Shaun Aug 09 '23 at 13:08
  • @peek-a-boo No offense, but all that is beyond my comprehension :-) Can you please suggest a reference aimed at physics students but at a higher level than Georgi's? Thanks :-) – Solidification Aug 09 '23 at 13:30
  • The stuff I said isn’t really specific to Lie groups/algebras. It is mainly differential geometry/topology stuff, and really (besides the jargon) multivariable calculus stuff related to the inverse/implicit function theorem and how that allows us to define (sub)manifolds in various ways. See here for the statements; at the end I also provide some references (the video lectures are nice:) See here for the proof of equivalence of the 4 conditions. – peek-a-boo Aug 09 '23 at 13:37
  • The point is that to have a ‘good parametrization’ $\Phi:\Bbb{R}^k\to M$ (e.g polar coordinates with $k=2$ or spherical/cylindrical coordinates with $k=3$) we need the map $\Phi$ to be injective (so different points in the parameter space get sent to different points in the manifold), and we need the tangent vectors $\frac{\partial\Phi}{\partial x^i}(p)$, for $1\leq i\leq k$ to be linearly independent (or what amounts to the same thing, $D\Phi_p$ needs to be injective linear map for all $p$). This says ‘infinitesimally we preserve the dimension’. Finally, a technical condition for $\Phi^{-1}$. – peek-a-boo Aug 09 '23 at 13:41

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