/sci/ - Science & Math

How do people do their change of basis calculations? I want something easy to remember but also generalises nicely to tensor fields when we are changing basis of tangent vectors and the transformations are by the Jacobian.

I have temporarily settled with pic related's convention, that is if we have old basis [math]e_i[/math] and new basis [math]e_i'[/math] the matrix [math]F[/math] taking old to new is obtained by expressing each new basis vector as a linear combination of old and stacking those coefficients into a matrix. Then all change of basis calculations are done by left multiplying [math]F[/math] by row vectors.

I like this because its nice for Jacobians, for example cartesian to polar we have tangents [math]e_x, e_y[/math] expressing new in old via chain rule gives [math]e_r = \frac{\partial x}{\partial r}\frac{\partial}{\partial x} + \frac{\partial y}{\partial r}\frac{\partial}{\partial y}[/math] and doing the same with angle gives

[math][e_x, e_y]\left[\begin{array}{cc}
{\frac{\partial x}{\partial r}} & {\frac{\partial x}{\partial \theta}} \\
{\frac{\partial y}{\partial r}} & {\frac{\partial y}{\partial \theta}}
\end{array}\right] = [e_r, e_\theta][/math]

When I try this with other methods I usually end up getting transpose Jacobians. I don't like this for just normal linear algebra though, its nasty having to write vectors of vectors and then do left multiplication. Is there a best of both worlds method?