Re: [Algorithms] decompose onto non-orthogonal vectors

[<ro...@do...>]

Discoe, Ben  wrote:

>
>> From: ro...@do... [mailto:ro...@do...]
>>
>> Chapter 1 in any linear algebra book.
>
>I assure i looked in chapters 1 though 18 of my book ("calculus and analytic
>geomtry", thomas/finney) and nothing resemling a formula for decomposing a
>vector is anywhere in it. 

Ah, while that is a fine calculus book, it is not an elementary linear
algebra book.  The main point you missed is that the single vector
equation is a set of two (in 2D) or three (in 3D) scalar equations.

> In general, math textbooks tend to be concerned
>with axioms and definitions, not with formulas that actually prove to be
>useful.
>

This assertion betrays an utterly mathophobic attitude, which is
likely to hinder your continuing progress.  The fact is that a lot of
the math nonsense we see posted to lists like this simply results from
people not taking sufficient care to define their terms.

>> That is A SYSTEM OF TWO LINEAR EQUATIONS IN TWO UNKNOWNS u, v,
>> which you ought to have learned to solve in intermediate algebra,
>
>I was asking if somebody knew the solution, not for how to do the math,
>which i'm willing to leave to people who enjoy re-deriving solutions to math
>problems.
>
>> 	u = (p1 b2 - p2 b1)/(a1 b2 - a2 b1)
>> 	v = (a1 p2 - a2 p1)/(a1 b2 - a2 b1)
>

Again a statement that doesn't make much sense to me.  You can't learn
in advance all solutions to all problems. Education consists of
learning how to have the courage to attack problems whose solution you
don't already know.  It would have been no benefit at all to you to to
simply write down the above formulas, in fact it would have been wrong
because, as it turns out, you were actually thinking of the 3D version
of the problem.

That's another important relevant point.  There is no hope of solving
a problem, or even having a sensible discussion about it, if it is not
clearly and completely stated.  An enormous number of the math queries
that we see posted to newsgroups and email lists are impossible to
answer because they are not clearly stated, the problem statement
doesn't make sense.  Taking math courses from good teachers is one of
the best ways to improve ones understanding of when a problem
statement makes sense.

>OK, i'll believe that after solving for v and substituting twice (a page or
>so of algebra) you get that 2d solution.
>



>> If the denominator (a1 b2 - a2 b1) is zero then it means that a and b
>> are linearly dependent (i.e. collinear) and either it has no solution
>> (if p is not also collinear with a and b) or infinitely many different
>> solutions (if p is collinear with a and b)
>
>The picture i drew showed a nondegenerate case.
>

Yes, but whenever one writes down an algebraic formula that involves
division, one MUST ask under what conditions the divisor can be zero.
For good reasons, computers don't like to divide by zero.  

>> Now suppose that you are given this as a 3D problem, say
>> 	a = (a1,  a2,  a3) and b = (b1, b2, b3)
>
>Yes.  I'm sorry i forgot to state that i do have 3d vectors, and p is known
>to lie in the plane formed by a and b.
>

Apology accepted. See comment above. 

>> It may or may no have solutions..
>> Cramer's rule does not apply here, but Gaussian elimination
>> still does.
>
>Is this your way of saying you don't know the answer?
>
OF COURSE NOT.  

>It would be pretty sad if not a single person on the list knows the answer.
>I was kinda hoping *you* would, Ron :)
>

The point you have evidently missed is that in the 3D case nobody
knows the answer a prior, nobody can write down a simple algebraic
formula giving the answer.    The point is that the algorithm for
finding the answer in ANY case is just one of the basic techniques
that we learn in elementary linear algebra. 
 
Talk about "sad":    What makes me sad is prevalence of mathophobia.
What makes me sadder is that so few participants in this list could
help you here.  What makes me saddest is the posting of completely
erroneous solutions to problems such as this.

>> No, I will not belabor the list with a review of Gaussian elimination
>> algorithms.  You will find one in Chapter 1 of your favorite linear
>> algebra book.
>
>The index in my textbook does not list "Gaussian elimination".
>

Try a linear algebra textbook, not a calculus and analytic geometry
text book.

>Thanks,

Welcome