Rotlx

Rotlx is a fictional mathematical operation used as a conceptual construct to study how a rotation interacts with a general linear transformation on vectors in n-dimensional space. It is defined as the composition v -> R L v, where R is a rotation matrix in SO(n) and L is an invertible linear transformation. The order of composition matters, and some sources also consider v -> L R v.

Properties of rotlx include that det(rotlx) = det(R) det(L) = det(L) since det(R) = 1 for a rotation. Therefore rotlx is invertible if both R and L are invertible. It is generally not an orthogonal transformation unless L itself is orthogonal. Rotlx is not a standard object in classical linear algebra but serves as a teaching and exploratory tool for examining how the order of transformations affects outcomes.

Example: In three dimensions, let R be a 45-degree rotation about the z-axis, and let L = diag(2, 1, 1) (scaling along the x-axis by 2). Then rotlx(v) = R L v scales the x-coordinate and then rotates, yielding a result that typically differs from applying L R v.

Applications and context: Rotlx is used in pedagogy to illustrate non-commutativity of transformation order, in reasoning about computer graphics pipelines, and in speculative discussions about the effects of composing rotations with general linear transformations. It is presented here as a fictional construct rather than a standard mathematical object.

See also: Rotation, Linear transformation, Matrix multiplication, Composition of functions.