Let $v_1 = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$ and let $v_2 = \begin{bmatrix} 1 \\ 3 \end{bmatrix}$. Let $x$ and $y$ be the coordinates with respect to the standard basis $e_1, e_2$ and let $u$ and $v$ be the the coordinates with respect to $v_1, v_2$. Write the equations to translate from $(x, y)$ to $(u, v)$ and back.
My answer for $(u, v)$ to $(x, y)$ is the following $$u+v=x$$ $$u+3v=y$$
My answer for the reverse is the following $$\frac{1}{2}(3x-y)=u$$ $$\frac{1}{2}(-x+y)=v$$
While this answer makes sense computationally, intuitively it feels weird. The first set of equations can obviously be described by the linear transformation represented by the matrix equation $$\begin{bmatrix} 1 && 1 \\ 1 && 3 \end{bmatrix} \begin{bmatrix} u \\ v \end{bmatrix} = \begin{bmatrix} x \\ y \end{bmatrix}$$ However, over here to try to understand the linear transformation intuitively I try to understand what is being transformed. The transformation is from $\mathbb{R}^2 \rightarrow \mathbb{R}^2$. What I am confused about is why is this a transformation from $(u, v)$ to $(x, y)$ and not the other way around.
When I thought about this question, I tried to compare it to a general linear transformation from $\mathbb{R}^3 \rightarrow \mathbb{R}^2$ to gain a better understanding. This transformation is represented by a 3 by 2 matrix. The columns are vectors in $\mathbb{R}^3$ that the standard basis vectors of $\mathbb{R}^2$ map to. Basically, the columns represent something about the output as evidenced by the fact that the columns have 3 entries in them. We take in a 2-dimensional vector and output a 3 dimensional vector and the 3-D vector is a linear combination of the two vectors in the columns. That is why we have a $3 \times 2$-matrix, that can be used to compute individual transformed vectors using matrix multiplication. I associated the 3 in the "3 by 2 matrix" with the codomain of $\mathbb{R}^3$ and the 2 in the "3 by 2 matrix"
I tried to apply this same sort of logic to the problem at hand. This much I understand that we are taking a two-dimensional vector in one coordinate system and putting it in another coordinate system. The columns of the matrix are basis vectors for the nonstandard coordinate system $(u, v)$. However, unlike the 3 by 2 matrix where we took in a 2-D vector with 2 coordinates, here we are taking in coordinates in the $(u, v)$ system and the columns appear to represent something about the input, not the output as in the 3 by 2 matrix. Can someone help me understand this supposed discrepancy?