Moving frame for a family of subspaces

A moving frame for a smooth family of $k$ -dimensional subspaces of $\mathbb R^n$ is a smooth matrix $B(q)\in\mathbb R^{n\times k}$ of full column rank whose columns span the subspace at parameter $q$ . A vector in that subspace is uniquely written $v=B(q)c$ ; a left inverse recovers $c=L(q)v$ .

Differentiating a frame representation

Along a differentiable parameter curve,

v'=B c'+B'c, \qquad c'=L(v'-B'c).

The term $B'c$ records the moving basis. If $v'=Kv$ and the evolution is compatible with the moving subspaces, then

c'=L(KB-B')c.

Choosing an orthonormal frame gives $L=B^T$ . A nonorthonormal frame instead needs its actual left inverse; the derivative terms persist in either choice.

References

Sheldon Axler, Linear Algebra Done Right, 4th ed., Chapters 2–3, 6 and 9.