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A convenient way to express a linear transformation is by use of homogeneous coordinates. If we write $z=z_1/z_2$ and $w=w_1/w_2$ we find that $w=Sz$ if $$w_1=az_1+bz_2\text{ and } w_2=cz_1+dz_2$$

All linear transformations form a group. The ratios $z_1:z_2\neq 0:0$ are the points of the complex projective line, and the two equations above identify the group of linear transformations with the one-dimensional projective group over the complex numbers, usually denoted by $P(1,\mathbb{C})$.

What is meant by the complex projective line here? What about the one-dimensional projective group over the complex numbers?

Isn't this transformation just taking the set of extended plane (i.e. all complex numbers, union with infinity) onto itself?

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Mika H.
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    See this Wikipedia-article for a definition of the projective line. Also observe the paragraph discussing the fact that you also observed: the projective line over the field of complex numbers is for all purposes just the Riemann sphere. The use of homogeneous coordinates makes it easier to see how invertible 2x2 matrices act on the projective line. Also homogeneous coordinates generalizes naturally to higher dimensions. They are used heavily in e.g. algebraic geometry. – Jyrki Lahtonen Sep 01 '13 at 09:05
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    "Complex projective line" is the same as "Riemann sphere", like "complex line" of an algebraic geometer is the same as the "complex lpane" of an analyst:-) This indeed creates some confusion sometimes. – Alexandre Eremenko Mar 29 '15 at 15:46

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