有人可以给出关于 sympy.vector.matrix_to_vector 的几何解释吗？

Question

SymPy doc gives an explanation about this function sympy.vector.matrix_to_vector

Converts a vector in matrix form to a Vector instance.

It is assumed that the elements of the Matrix represent the measure numbers of the components of the vector along basis vectors of 'system'.

and this example

>>> from sympy import ImmutableMatrix as Matrix
>>> m = Matrix([1, 2, 3])
>>> from sympy.vector import CoordSys3D, matrix_to_vector
>>> C = CoordSys3D('C')
>>> v = matrix_to_vector(m, C)

It seems that C represents the coordinate axes of a 3d Euclidean space.

As per standard convention, Ci, Cj, Ck represent basis vectors along the 𝐗, 𝐘 and 𝐙 axes respectively.

It seems that m = Matrix([1, 2, 3]) is the Matrix we're considering (the matrix to be converted to a vector). Can someone give a geometric interpretation about converting the vector in matrix m to Vector instance ?

Is there a point (1, 2, 3) somewhere in that space? What is it for?

Answer 1

There are actually a lot of implicit assumptions behind " a point (1,2,3) in space ". When we use coordinates such as (1,2,3) what we really mean is a representation (a specific thing) of an abstract object (an element of a set; the set is a vector space ℝ³ and an element is called a vector).

The way we represent vectors in vector spaces is that we first specify a basis and then use the guaranteed-to-be-real numbers which define a unique linear combination of the basis vectors that specify a given vector in the vector space. When we colloquially say " a point (1,2,3) " we imply

• the linear space (vector space) is ℝ³
• the basis is the canonical Cartesian basis i , j , k which is an orthonormal basis .

So coming back to your question: the "vector in matrix form" [1, 2, 3] can only be made sense of if we have a good understanding of the vector space and the basis, especially the latter. If our basis is the Cartesian basis then this is the point 1i + 2j + 3k which has a distance of sqrt(14) from the origin. If our basis was something else (and we can have infinitely many valid bases in a given vector space), say, I = 2i - 3j , J = -k + i and K = j + 2k then on this basis the same coordinates (1,2,3) (note that I used parentheses instead of square brackets to indicate the change in basis) would correspond to the point 1I + 2J + 3K = (2i-3j) + 2(-k + i) + 3(j + 2k) = 4i + 0j + 4k which has a distance of sqrt(32) from the origin.

So the core of the problem is that coordinates are tied to a basis, and the "matrix form" of the vector can only contain coordinates. But using sympy.vector you can use abstract vectors as first-class objects, in the doc's example the matrix [1,2,3] gets converted to Ci + 2*Cj + 3*Ck which is a well-defined element of a well-defined linear space. If you use a different basis to do the conversion you will get a different (but again well-defined and abstract) element of the vector space.

And the final step (ie the geometric interpretation) is that elements of the vector space ℝ³ correspond to what we think of as points in 3d real space. So yes, in the example you showed from the docs this transformation from "matrix form" to vector is a bit hard to grasp, because there's only a subtle change from concrete representation to abstract. Choosing different bases for the conversion might make it more obvious that what we call [1,2,3] should not be taken for granted.

Answer 2

The function sympy.vector.matrix_to_vector gives you a vector v from a given matrix m .

Under your assumption, Ci, Cj, Ck represent basis vectors along the 𝐗, 𝐘 and 𝐙 axes respectively in a 3d Euclidean space, v is a vector that starts from the origin and points toward the point (1, 2, 3) .