Change of basis for linear maps and bilinear forms

dear professor,

when I read the note, I find these problems, hope you do not mind

1. in page 40, at the end of 4.1.100 proposition, you wrote “and for any linear map T:V \rightarrow V we have \left[T\right]_C= M^{-1}\left[T\right]_B M,” can you explain why? because in 4.1.101, the note said \left[f\right]_C = M^{t}\left[f\right]_B M, one is M^{-1} and one is M^{t} which makes me a little confused.

2. in page 40, I think all the appearance of “theorem 5.1” should be changed to “theorem 4.1”. am I right?

thanks very much


1. The intention of that paragraph was exactly to warn you that the way the matrix of a bilinear form changes is different from the way the matrix of a linear map changes when we go from one basis to another.

2. Yes, it should have been theorem 4.1. Thanks very much for the correction.


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