How get vectors in high-dimensional space?

Hello. I have distances matrix. What method I must use to map vectors to high-dimensional space?

This question is quite vague and so hard to answer.
Perhaps provide more specifics ?... a simple example ?

Can you be more explicit about your use case ?

Ok. I need to get coefficients of hyperplane from the matrix. If I have ND space I have W1, W2, ..., Wn, and B coefficients.
So, If I want to find these coefficients I need to compute the determinant of matrix NxN, if I have a submatrix where all elements are the same then the determinant is zero.

Can you illustrate the issue with code?

The main idea is Wk = det(matrixK), matrixK = original matrix without column of K and add column 1.
Example:
Matrix :
[0.2 0.4 0.1 0.6]
[0.9 0.1 0.5 0.2]
[0.7 0.3 0.6 0.3]
[0.5 0.7 0.3 0.1]
W1 = det (Matrix1) as
det
(
[0.4 0.1 0.6 1.0]
[0.1 0.5 0.2 1.0]
[0.3 0.6 0.3 1.0]
[0.7 0.3 0.1 1.0]
)

The same way for the next issues....

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Ok. I have a distance matrix as Euclidean distances between points in 2D space. Now I need to get the same points but in high-dimensional space (4D, 5D, 6D...). How I can do that?

mmmm.....

what do you think of this ?

#this might represent x=1 , y=2
(m1 <- matrix(c(1,2)))

# addng a third dimension (z?) with value 3
(m2 <- rbind(m1,3))

P2d1 = {x1, y1} and P2d2 = {x2, y2} - > distance2
P5d = {x1, y1, 0,0,0} and P5d = {x2, y2, 0,0,0} -> distance2
also
P5d = {x1, y1, 1,1,1} and P5d = {x2, y2, 1,1,1} -> distance2
there are infinity solutions

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Nope. It must be the same distances between points in new space (Or try to find the closest variant).
P2d1 = {x1, y1} and P2d2 = {x2, y2} - > distance2
P5d = {x1, y1, z1, k1, r1} and P5d = {x2, y2, z2, k2, r2} -> distance5
So, these 2 distances must be very close.
Yep, I can use Bruteforce =), but I have many points, so I need another method for this.

P5d = {x1, y1, 1,1,1} and P5d = {x2, y2, 1,1,1} -> distance2
there are infinity solutions

It's a beautiful solution, I'll try this in my test case.

It isn't work in my case, because I have zero determinant in the submatrix.
I need a real method to find points.
I heard about MDS for low-dimensional space, but I don't know any methods for high-dimensional space.
MDS doesn't work in my case.