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Does change of basis matrix we use in linear transformations change both the domain and range of the transformation matrix?

By the way, I have a hard time calculating the change of basis matrix. I've been looking up all the internet But couldn't find the best method that fits me. If you can enlighten me in that part, that would be great :)

Thanks

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    Change of basis matrices $I_{\mathcal C\leftarrow \mathcal B}$ are actually constructed to act on vectors. But due to associativity: $$I_{\mathcal B\leftarrow \mathcal C}(M_{\mathcal C\leftarrow \mathcal C}(I_{\mathcal C\leftarrow \mathcal B}[v]{\mathcal B})) = (I{\mathcal B\leftarrow \mathcal C}M_{\mathcal C\leftarrow \mathcal C}I_{\mathcal C\leftarrow \mathcal B})[v]{\mathcal B} = M{\mathcal B\leftarrow\mathcal B}[v]_{\mathcal B}$$ Note that if you only applied a change of basis matrix to one side of a matrix you'd only change the basis representation of the domain or codomain. –  Jul 20 '16 at 20:00
  • To learn about change of basis, try: your textbook, Khan Academy, my answer to this question, my answer 2, or my answer 3. –  Jul 21 '16 at 02:57

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