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Anonymous
How can I prove that differential is invariant as linear map 2017-06-10 11:33:27 Post No. 329269
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How can I prove that differential is invariant as linear map 2017-06-10 11:33:27 Post No. 329269
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How can I prove that differential is invariant as linear map
Anonymous
2017-06-10 11:33:27
Post No. 329269
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I'm stuck on the problem in my linear algebra task. I've spent the whole fucking day with it and can't figure out which way to go.
Let f: R^m -> R^n be the mapping defined by differentiable functions f_i, which are, in general, nonlinear and map zero to zero:
F(x_1, ..., x_m) = (..., f_i (x_1, ..., x_m), ...), i = { 1, ..., n}, f_i (0, ..., 0) = 0.
A linear map df_0: R ^ m -> R ^ n, is called the differential of f at the point 0 and given by the formula on OP pic. {e_j}, {(e_i)'} are standard bases in R^m and R^n, respectively. I must show that if I replace the bases in both spaces and compute df_0 by the same formulas in other bases, then the new linear map df_0 coincides with the old one.
How can I possibly solve this?
>>
The thing to understand is that the x_j are the coordinate functions on R^m belonging to the basis e_j. When you choose a different basis, you get different coordinate functions!
So when you change the basis in R^m, you have to use the chain rule.
(Similarly, {f_i} are the coordinates of the vector-valued function f, expressed with respect to the basis {e'_i} of R^n. No chain rule needed here, just linearity of derivatives.)
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