# Vector fields generating a transformation

It would be great if someone can explain to me what the following means:

Vector fields $V_i, i=1,2,3$ generate 3 single-parameter groups of transformations in $\mathbb R$ -- $$\tilde x =x\exp(\alpha);$$$$\tilde x = x+\beta;$$$$\tilde x = {x\over 1-\gamma x}$$ respectively. These 3 vector fields together generate a 3-parameter group of transformations: $\tilde x = {ax+b\over cx+d}$ where $ad-bc=1$

I think I understand the first bit, and I think the $V_i$'s are $x\partial_x; \partial_x;x^2\partial _x$ respectively. However, I have no clue what the second bit is talking about -- how do you combine them and why is that so?

Qiaochu has helpfully pointed out that this has to do with Lie algebra. Could anyone further explain how they combine these vector fields/transformations to give the 3-parameter transformation group?

Thank you.

Anyone?

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The second bit asserts that three vector fields span a Lie algebra (en.wikipedia.org/wiki/Lie_algebra) which generates a Lie group (en.wikipedia.org/wiki/Lie_group). –  Qiaochu Yuan Nov 24 '12 at 2:18
Thank you, @QiaochuYuan , how though did they combine them to get that resulting transformation? –  Henry Nov 24 '12 at 2:26