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I am reading section 46 of Halmos' Finite Dimensional Vector Spaces. In this section Halmos poses two questions the first of which is:

Question 1: If $x$ is in a finite dimensional vector space $V$, write $x = \sum_i \xi_i x_i = \sum_i \eta_i y_i$, what is the relation between its coordinates $(\xi_1,\xi_2, \ldots \xi_n)$ with respect to the basis $X = (x_1, \ldots, x_n)$ and its coordinates $(\eta_1, \ldots ,\eta_n)$ in the basis $Y = (y_1 \ldots y_n)$?

After a few lines, Halmos defines the linear transformation $A$ by $A(x_i) = y_i$, $i=1,2, \ldots n$. From what I understand, suppose we have a basis vector $x_i$. Then this should correpond to the column vector

$\left[\begin{array}{c} 0 \\ 0 \\ \vdots \\ 1 \\ \vdots \\ 0 \end{array}\right]$

in the basis $X$ where the $1$ is in the $i-th$ row of the vector.

Then if the matrix $A = a_{ij}$ is applied to this vector, we should have the $i-th$ column of the matrix. Let the $i-th$ column of the matrix be

$\left[\begin{array}{c} a_{1i} \\ a_{2i} \\ \vdots \\ a_{ni} \end{array} \right].$

However note that the $i-th$ column of the matrix is expressed in terms of the basis $X$, because Halmos states that $y_j = Ax_j = \sum_i a_{ij}x_i$. In other words, he has chosen to express each vector in the basis $Y$ as a linear combination of the basis vectors in $X$. Is there a particular reason of doing this? From my limited experience in linear algebra, if say we have a vector in a basis $X$, and we wish to express it in a basis $Y$, the thing to do would be to express $x_1$ as a linear combination of the y_i's in $Y$, $x_2$, and so on. This is the opposite to what Halmos has done.

Is this difference anything significant to be aware of?

$\textbf{Edit : }$ I will write out the relevant bit that I am referring to in Halmos' book:

"Let $V$ be an $n$- dimensional vector space and let $X = (x_1, \ldots x_n)$ and $y=(y_1,\ldots y_n)$ be two bases in $V$. We may ask the following two questions:

Question 1. If $x$ is in $V$, $x = \sum_i \xi_ix_i = \sum_i\eta_i y_i$, what is the reation between its coordinates $xi_i$ with respect to $X$ and its coordinates $\eta_i$ with respect to $y$?

Question 2. If $(\xi_1, \ldots \xi_n)$ is an ordered set of $n$ scalars, what is the relation between the vectors $x = \sum_i \xi_ix_i$ and $y = \sum_i \xi_i y_i$?

Both these questions are easily answered in the language of linear transformations. We consider, namely the linear transformation $A$ defined by $Ax_i = y_i$. More explicitly:

$A(\sum_i \xi_i x_i) = \sum_i \xi_iy_i.$ "

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    Merci, je me suis rendu compte que ce fait est très simple! Au fait si vous voyez le résultat à la fin de sa question (Question I) il a dit: $\xi_j = \sum_i a_{ij} y_i$ qui est quelque chose de bien connu!2011-08-15

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The matrix which expresses basis vectors of one basis $X$ in terms of another basis $Y$ and the matrix which expresses vectors of $Y$ in terms of $X$ are just inverses to each other and which one you call $A$ and which one becomes $A^{-1}$ is a matter of notation or convention - you just have to pick the right one when doing an actual change of coordinates.