Find representation of a 3d vector as a combination of 2 basis vectors using python

Question

I want to find the representation of vector (-9, 1, -9/2) in basis B = {v1, v2} = {(10, 4, 5), (2, 1, 1)} using python. I know that the answer is (-11/2, 23) , this question is on how to find that result using python. I'm not interested in writing the algorithm from scratch, but instead am wondering what methods from numpy / similar libraries I should use.

Given the following matrix:

A = np.array([ [10,2,-9], [4, 1, 1], [5, 1, -9/2]])
array([[10. ,  2. , -9. ],
       [ 4. ,  1. ,  1. ],
       [ 5. ,  1. , -4.5]])

The result should be

array([[ 1  ,     , -11/2 ],
       [    ,  1  ,  23 ],
       [    ,     ,   0]])

Note - I've typed the result out there, I'm not necessarily expecting results represented as fractions using sympy or such.

If this were to be computed by hand, the steps might be as follows:

Step 1

10     2       -9
4      1       1
5      1       -9/2

Step 2

1      1/5     -9/10
4      1       1
5      1       -9/2

Step 3

1      1/5     -9/10
0      1/5     4.6
0      0       0

Step 4

1      0       -5.5
0      1       23
0      0       0

Edit

The following may be used:

In [2]: A = np.array([ [10,2,-9], [4, 1, 1], [5, 1, -9/2]])
In [5]: sympy.Matrix( A ).rref()
Out[5]:
(Matrix([
 [1, 0, -5.5],
 [0, 1, 23.0],
 [0, 0,    0]]),
 (0, 1))

Somewhat surprised there's no solution using NumPy.

Yes, but then you'd have to guarantee that the third vector is a linear combination of the first two (tip: the matrix from these 3 vectors should have 0 determinant). — Andras Deak -- Слава Україні, Mar 21 '20 at 14:00
@AndrasDeak well, yes I guess that's exactly what this is? We're expressing x in terms of B, so it's reduced from a vector in R^3 to a vector in R^2 that's within the plane created by v1, v2 — baxx, Mar 21 '20 at 14:02
Your steps show Gaussian elimination of a matrix (without pivoting). That's what the suggested dupe asks for. — Andras Deak -- Слава Україні, Mar 21 '20 at 14:14
@AndrasDeak It seems that there's not a way of doing this in numpy then? I didn't really want to / expect to have to use sympy. — baxx, Mar 21 '20 at 14:22
And the reason there's no built-in for this might be that the algorithm is inaccurate. In most applications you could use LU decomposition, but that probably involves pivoting. — Andras Deak -- Слава Україні, Mar 21 '20 at 14:23
@AndrasDeak I was interested in a numpy solution - are you saying that this isn't easily obtained? — baxx, Mar 21 '20 at 14:24
@roganjosh it seems that might have to be used - should I add the example from this post to that thread? As I feel it would be useful for others — baxx, Mar 21 '20 at 14:29
An answer here, http://numpy-discussion.10968.n7.nabble.com/Reduced-row-echelon-form-td16486.html claims RREF more of a textbook linear algebra tool, and not that useful or easy in numerical linear algebra. — hpaulj, Mar 21 '20 at 15:31
For your specific use case you can use the underdetermined system of equations, `A = np.array([[10, 2], [4, 1], [5, 1]]); b = np.array([-9, 1, -9/2]); x = np.linalg.pinv(A) @ b`. Can't vouch for the stability of this method, and you should check first for singularity of your original `A` matrix. — Andras Deak -- Слава Україні, Mar 21 '20 at 15:32

Find representation of a 3d vector as a combination of 2 basis vectors using python

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