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This question already has an answer here:

For example if i have a linear operator $\ T:\mathbb{R}^n\to \mathbb{R}^n$ and some basis $\ B=\{v_1,v_2,..v_n\}$.

Let $\ [T]_B$ be the representation matrix of $\ T $ with respect to the basis $\ B$. for example if I do $\ [T]_B*v_2 $ do I get the coordinates of $\ v_2$ in the standard basis?

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marked as duplicate by Fabian, Community May 1 '17 at 12:57

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ How one can call this question an "exact duplicate" (later mollified to "duplicate") of the linked earlier question is beyond me. I admit that in both questions the word "matrix" occurs. The OP here has no clear idea what the matrix of a linear transformation expresses. He is definitely not talking about the way this matrix changes under a basis change. $\endgroup$ – Christian Blatter May 1 '17 at 17:52
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Let's take the standard basis to be $E$ and another basis to be $B$. Then you could find the Basis Change Matrix $Q$ with the help of the transformation which will eastablish the connection between the bases $E$ and $B$. $Q$ will change the co-ordinates of a vector w.r.t $E$ to the co-ordinates of that same vector w.r.t $B$.

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