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I am spending time revising vector spaces. I am using Dummit and Foote: Abstract Algebra (Chapter 11) and also the book Linear Algebra by Stephen Freidberg, Arnold Insel and Lawrence Spence.

On page 419 D&F define similar matrices as follows:View attachment 3047

They then state the following:View attachment 3048

BUT? ... how exactly does it follow that \(\displaystyle P^{-1} M_{\mathcal{B}}^{\mathcal{B}} (\phi) P = M_{\mathcal{E}}^{\mathcal{E}} (\phi)\)

Can anyone show me how this result is obtained?

I must be missing something obvious because no indication is given of how this result is obtained ... ... ?

Peter

***EDIT***

(1) Reflecting ... ... I am beginning to think that \(\displaystyle M_{\mathcal{B}}^{\mathcal{B}} (\phi)\) and \(\displaystyle M_{\mathcal{E}}^{\mathcal{E}} (\phi)\) are equal to the identity matrix \(\displaystyle I \) ... ... ? ... ... but then, what is the point of essentially writing \(\displaystyle P^{-1} I P = I\)?(2) Further reflecting ... ... It may be that the above formula makes more sense in the overall context of what D&F say about the change of basis or transition matrix ... ?

In the light of (2) I am proving the relevant test on similar matrices and the transition of change of basis matrix for MHB members interested in the post ... ... see below ... ... View attachment 3049

On page 419 D&F define similar matrices as follows:View attachment 3047

They then state the following:View attachment 3048

BUT? ... how exactly does it follow that \(\displaystyle P^{-1} M_{\mathcal{B}}^{\mathcal{B}} (\phi) P = M_{\mathcal{E}}^{\mathcal{E}} (\phi)\)

Can anyone show me how this result is obtained?

I must be missing something obvious because no indication is given of how this result is obtained ... ... ?

Peter

***EDIT***

(1) Reflecting ... ... I am beginning to think that \(\displaystyle M_{\mathcal{B}}^{\mathcal{B}} (\phi)\) and \(\displaystyle M_{\mathcal{E}}^{\mathcal{E}} (\phi)\) are equal to the identity matrix \(\displaystyle I \) ... ... ? ... ... but then, what is the point of essentially writing \(\displaystyle P^{-1} I P = I\)?(2) Further reflecting ... ... It may be that the above formula makes more sense in the overall context of what D&F say about the change of basis or transition matrix ... ?

In the light of (2) I am proving the relevant test on similar matrices and the transition of change of basis matrix for MHB members interested in the post ... ... see below ... ... View attachment 3049

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