Change of basis matrix from $\alpha$ to $\beta$ or from $\beta$ to $\alpha$?

Question

In Peterson and Sochacki's Linear Algebra and Differential Equations they define (in section 5.3) the change of basis matrix from $\alpha$ to $\beta$, $[I]_{\beta}^{\alpha}$, as the matrix whose columns are the $\alpha$-coordinates of the $\beta$ vectors. This matrix transforms $\beta$-coordinates into $\alpha$ coordinates so I think it should be called the change of basis matrix from $\beta$ to $\alpha$.

Is there a good reason to call it the change of basis matrix from $\alpha$ to $\beta$?

Answer 1

Yes, there is an excellent reason: you can obtain the matrix $A'$ of the associated linear map in basis $\beta$ from the matrix $A$ of this linear map in basis $\alpha$.

Indeed, let's denote $X, Y,\dots$ the column matrix of vectors in basis $\alpha$ and $X', Y',\dots,\:$ their column matrix in basis $\beta$. If $Y $ is the column matrix of coordinates of the image of a vector with column vector of coordinates $X$ in basis $\alpha$, we have the relation $$Y=AX.\tag1$$ Now, denoting $P$ the change of basis matrix, the column matrices of the same vectors, in bases $\alpha$ and $\beta$ are linked through the relations $$X=PY',\qquad Y=PY',$$ so that $(1)$ can be written as $$PY'=A(PX')\iff Y'=P^{-1}A(PX')$$ which shows that the matrix of the linear map in basis $\beta$ has become $$A'= P^{-1}AP.$$

Answer 2

One reason could be that "coordinates" are duals to "vectors" themselves.

While the matrix transforms $\beta$-coordinates into $\alpha$-coordinates, it transforms $\alpha$-vectors to $\beta$-vectors via $$ (\beta_1,\ldots,\beta_n) = (\alpha_1,\ldots, \alpha_n) [I]. $$