Find change of basis matrix I'm asked to find the change of basis matrix from basis $\underline{e}$ to $\underline{f}$ given the following information:
The coordinate relationship is given by:
$$3y_1 = -x_1 + 4x_2 + x_3$$
$$3y_2 = 2x_1 + x_2 + x_3$$
$$3y_3 = 0 + -3x_2 + 0$$
The coordinates $x_i$ belongs to basis $\underline{e}$.
We know that $\underline{e}X_e = \underline{f}AX_e=\underline{f}X_f$ where $X$ is the coordinates in the basis given by the subscript.
$\underline{f}X_f=\underline{f}AX_e$ is exactly what the coordinate relationships say. So to me the transformation matrix is given by:
$$A =\begin{pmatrix}
-1 & 4 & 1\\ 
2 & 1 & 1\\ 
0 & -3 & 0
\end{pmatrix}$$
But the answer is the inverse of that.
Can someone explain where my logic fails?
 A: I will explain change of basis with a simpler example, perhaps it is easier to see how this inverting comes into being. This may be a bit easier to follow if you accept the philosophy that vectors exist in the space regardless of any bases or coordinate grids. When we choose a basis in the space, the vectors get an algebraic representation given by coordinates. When we're changing basis, we're not changing the vector, but rather the coordinate grid around the vector. At the same time, we do want the transition matrix to act on the coordinates of the vector. It is this that gives rise to the inversion, since algebraically, instead of stretching the grid (which is the geometric interpretation), we're shrinking the vector.
Say we have our space $\Bbb R^3$, and we have two bases $\underline g$ with basis vectors $x_1, x_2, x_3$ and $ \underline h$ with basis vectors $y_1, y_2, y_3$. Since this is a simple example I will just assume that $\underline g$ is using feet and $\underline h$ is using meters, and that their axes agree. That means (with some rounding) that
\begin{align}
y_1 &= 3x_1 + 0x_2 + 0x_3\\
y_2 &= 0x_1 + 3x_2 + 0x_3\tag{*}\\
y_3 &= 0x_1 + 0x_2 + 3x_3
\end{align}
But what is the transition matrix from $\underline g$ to $\underline h$? Say you have a vector $X$, and its representation in the $\underline g$-basis is $X_g = (6, 9, 3)^T$. That means that from the origin we go $6$ feet in the first direction, $9$ feet in the second direction and $3$ feet in the last direction. Converting to meters, this is the same as going $2$ meters in the first direction, $3$ meters in the second direction and $1$ meter in the third direction, which is to say that the representation of $X$ in the $\underline h$-basis is $X_h = (2, 3, 1)^T$.
Writing this in matrix form, if $A$ is the transition matrix from $\underline g$ to $\underline h$, that is to say $X_h = AX_g$ for all vectors $X$, then $A$ needs to divide every entry by $3$, which means that we have
$$
A = 
\begin{pmatrix}
\frac13&0&0\\
0&\frac13&0\\
0&0&\frac13
\end{pmatrix}
$$
which is the inverse of the coefficient matrix of $\text{(*)}$ above.
