Re: [Algorithms] decompose onto non-orthogonal vectors

[Will P. <wi...@cs...>]

> how to you get scalars (u,v) such that u*a + v*b = p?
> Ie. decompose p onto a and b.

Regardless of what the answer is, you have one equation (ua + vb = p) and
two unknowns (u and v).  You need to come up with an equation involving u
and/or v that is linearly independent of the first equation, and then you
can solve the system of linear equations.

You could make up something like:

u + v = 1, which is the same as v = 1 - u

and then substitute in to the first equation:

ua + vb = p

ua + (1 - u) b = p

ua - ub = p - b

u = (p - b) / (a - b)

and then v = 1 - u... assuming I didn't make any mistakes in this textual
algebra, which is hard for me to write (and I imagine hard to read).

the thing is, this made-up equation (u + v = 1) doesn't guarrantee a
positive u and positive v.  so I hope you weren't looking for that. :)

the other problem is you have to make sure vector a and vector b are
linearly independent, or there is likely to be no solution, no matter what
u, v, and the second equation is.

Will