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In almols every book the topic of "change of basis" starts with such a paragraph:

Convince yourself that the effect of transformation $ T$ on the basis vectors $ \{\boldsymbol{e}_i\}$ can be described in matrix notation by $\displaystyle [\boldsymbol{e}_1, \boldsymbol{e}_2, \ldots, \boldsymbol{e}_n]\boldsymbol{A}=[A\boldsymbol{e}_1, A\boldsymbol{e}_2, \ldots, A\boldsymbol{e}_n],$

where the matrix $\displaystyle \boldsymbol{A}= \begin{bmatrix} a_{11}&a_{12}&\ldots\\ a_{21}&a_{22}&\ldots\\ \ldots \end{bmatrix}$

acts from the right on a row matrix. Notice that this is quite different from our usual appoach, where a vector becomes a whole column matrix rather than a single element of a row matrix. However, this notation is often handy as a mnemonic for how to construct the matrix that represents a transformation, and it serves as a warning not to construct column matrices whose elements are basis vectors.

Ok, so what just happend? Had author suddenly went to Modules over Rings? If so how then he explains that A, as a function on Module, being trasposed becomes a function on vector space of coordinate vectors?

Or is this a "lazy notation"? and instead of $\displaystyle [\boldsymbol{e}_1, \boldsymbol{e}_2, \ldots, \boldsymbol{e}_n]\boldsymbol{A}$ author should have written all this as a system of vector equations? as in the case of $e'_i=A{_i,_j}e_j$ ?

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  • $\begingroup$ It should say $$\begin{bmatrix}e_{1}\\ \vdots \\ e_{n}\end{bmatrix}A = \begin{bmatrix}e_{1}A \\ \vdots \\ e_{n}A \end{bmatrix}.$$ $\endgroup$ – Morgan Rodgers Jul 11 '16 at 13:21
  • $\begingroup$ And definitely the entire point is to write it as a matrix equation. $\endgroup$ – Morgan Rodgers Jul 11 '16 at 13:24
  • $\begingroup$ Maybe. Well, that's the reason I ask. I just don't understand this notation. I can undestand when we talk about "coordinate vectors" as columns and rows as functionals on colums. But this... stuff with vector as elements of a vector... I don't even know how this treatment of basis vectors is called, so I can't look up the details about "should matrix go befor or after vector or should the "basises vector" be row or a column... $\endgroup$ – coobit Jul 11 '16 at 13:27
  • $\begingroup$ The notation is, on the left it's a matrix whose $i$th row is (the row vector) $e_{i}$. And on the right, it's a matrix whose $i$th row is $e_{i}A$. The matrix goes on the right because it says the matrix "acts from the right". Also it has to be consistent; if the matrix is on the right on the left hand side of the equation, it has to also be on the right in the right hand side. $\endgroup$ – Morgan Rodgers Jul 11 '16 at 13:52
  • $\begingroup$ Did you copy this specific paragraph from a book? $\endgroup$ – Morgan Rodgers Jul 11 '16 at 13:55

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