torus is a surface having genus one

via Torus — from Wolfram MathWorld

is there a possibility to “transform” a matrix (inheriting the input-output “weights”) into a graph object inheriting the matrix relations?

you can convert it to an edge-weighted graph using `WeightedAdjacencyGraph`

. This will give you a complete graph (a `Graph`

expression) in which each vertex is also connected *to itself*. I am not sure how much sense it makes treat this matrix as a graph, given the full connectivity.

Since you have a fully connected graph, it does not matter in your case, but be aware that `WeightedAdjacencyGraph`

represents missing edges with `Infinity`

, not with zero. This is somewhat annoying because it is inconsistent with `WeightedAdjacencyMatrix`

, which uses `0`

. While `Infinity`

does make sense in *some* (not all) applications, it is much more common to see data that uses 0.

via [✓] Convert matrix into “Graph object”? – Online Technical Discussion Groups—Wolfram Community

Rotations

Source: Geometry of Linear Transformations of the Plane – HMC Calculus Tutorial

If V=R2 and W=R2, then T:R2R2 is a linear transformation if and only if there exists a 22 matrix A such that T(v)=Av for all vR2. Matrix A is called the standard matrix for T. The columns of A are T01 and T10 , respectively. Since each linear transformation of the plane has a unique standard matrix, we will identify linear transformations of the plane by their standard matrices. It can be shown that if A is invertible, then the linear transformation defined by A maps parollelograms to parallelograms. We will often illustrate the action of a linear transformation T:R2R2 by looking at the image of a unit square under T

Source: Geometry of Linear Transformations of the Plane – HMC Calculus Tutorial

the image T(P) of any point P is determined by noting that T(A) = A′, T applied to the line segment AB is A′B′, T applied to the line segment AC is A′C′, and T respects scalar multiples of vectors based at A. [If A, E, F are collinear then the ratio length(AF)/length(AE) is equal to length(A′F′)/length(A′E′).] Geometrically T transforms the grid based on ABCD to that based in A′B′C′D′.Affine transformations do not respect lengths or angles; they multiply area by a constant factor

In some cases, however, you may prefer to convert the system of linear equations into a matrix equation, and then apply matrix manipulation operations to solve it. This approach is often useful when the system of equations arises as part of a general algorithm, and you do not know in advance how many variables will be involved.

Source: Solving Linear Systems—Wolfram Language Documentation