**Problems 3 and 4: More complex equations**

The same methods may be used for more complicated equations. For
instance, suppose our measured quantities *y*_{i} depend
upon three different *known* variables:

leads eventually to

(Prove it!) These matrices have a beautifully simple and symmetric structure. I think that if you stare at them for a minute or two, you'll be able to memorize their form and recall them whenever you need them hereafter.

Notice that the known quantities *t*_{1},
*t*_{2}, and *t*_{3} do not need to be
independent of each other. For instance, if *t*_{1}
*x*^{2},
*t*_{2} *x*,
and *t*_{3} 1
then the coefficients of the best-fitting *parabola*

are easily seen to be

Or if

you can substitute

which gives

See? It's not called "linear" least squares because all it can do is
fit straight lines - it can fit all sorts of equations, provided they
are linear in the unknown parameters *a, b, c*, . . .