Two definitions of linearly independent vector fields I'm studying the chapter of Lee's ISM on vector fields, and there's one thing that confuses me a lot. If $\mathfrak{X}(M)$ denotes the set of smooth vector fields on a smooth $n$-manifold, then it is a vector space under pointwise vector addition and scalar multiplication:
$$(aX+bY)_p=aX_p+bY_p,\quad X,Y\in\mathfrak{X}(M).$$
Since $\mathfrak{X}(M)$ is a vector space, we can introduce the notion of linearly independence:
$$a_1 X_1+...+a_k X_k=0\Rightarrow a_1=...=a_k=0.$$
This is common material that can be found in any Linear Algebra course and is familiar to me. But in a while Professor Lee brought up an idea that seems like the one presented above. He said that an ordered $k$-tuple $(X_1,...,X_k)$ of vector fields defined on a set $A\subseteq M$ is called linearly independent if $(X_1|_p,...,X_k|_p)$ is linearly independent in $T_p M$ for each $p\in A$. Are these two ideas talking about the same thing? More precisely, are theses two statements equivalent if we consider $A$ to be all of $M$? Thank you.

Actually, the notion of linear independence of vector fields that I defined in ISM is not equivalent either to linear independence over $\mathbb R$ (viewing $\mathfrak X(M)$ as a vector space over $\mathbb R$), or to linear independence over $C^\infty(M)$ (viewing $\mathfrak X(M)$ as a module over $C^\infty(M)$). It's a separate thing, and stronger than either of the above notions of linear independence.
For example, in $\mathbb R^2$, the vector fields $\partial/\partial x$ and $x\,\partial/\partial y$ are linearly independent over $\mathbb R$ and also over $C^\infty(M)$. But they're not linearly independent vector fields in the sense I defined, because their values at points on the $y$-axis are not linearly independent.
Arguably, calling this concept "linearly independent vector fields" is a poor choice of terminology, just because of this confusion. It probably would have been better if I had called it "pointwise linearly independent." Something to put on my to-do list for a third edition, if there ever is one.
