owen by the way

May 27, 2011 in Cool Tricks, Exercises, Notations

composition of linear fractional transformations
compared to two-by-two matrix multiplications.

consider

in other words,
let f(x) = (Ax+B)/(Cx+D) and
let g(x) =(ax+b)/(cx+d) and
consider the function (“f\circ g”,
i.e. f-composed-with-g). recall
(or trust me on this) that
[f\circ g](x) = f(g(x)); i.e.,
functions compose right-to-left
(“first do gee to ex; then plug in
the answer and do eff *to* gee-of-ex”…
first g, then f… alas. but there it is).

so we have




thus

whereas one also has

so the matrix-multiplication equation
can be obtained from the function-composition equation
merely by applying an eraser here and there.

(my lecture-note-blogging of winter 09 include some
remarks on \mapsto notation and much more
about linear fractional (“mobius”) transformations.)

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