We define the coordinate vector as follows: given a smooth function , we define
Okay, I know that that’s confusing. But all we mean is this: start with a function . We compose it with the inverse of the coordinate map to get , where is some open neighborhood of the point . Now we can take that th partial derivative of this function and evaluate it at the point .
The first thing we really should check is that it doesn’t matter which representative we pick. That is, if in some neighborhood of , do we get the same answer? Indeed, in that case in some neighborhood of , and so their partial derivatives are identical. Thus this operation only depends on the germ .
But is it a tangent vector? It’s easy to see that it’s a linear functional, so we just have to check that it’s a derivation at :
And so we have at least these vectors at each point . We can even tell that they much be distinct — and even linearly independent — since we can calculate
where is the th coordinate projection . But we know that is always and everywhere — it takes the value if and otherwise.
Thus takes a different value on than on all the other . Further, any linear combination of the for must take the value on , while takes the value ; we see that none of the coordinate vectors can be written as a linear combination of the rest, and conclude that the dimension of is at least .