Is every linear transformation also technically a change of basis matrix? I don't really see the difference between a linear transformation and a change of basis matrix.
Let's take this formulation:
$$ Ax = v$$
I can interpret it in two different ways.
1.) The matrix $A$ encodes a linear transformation that converts vector $x$ to vector $v$.
2.) The matrix $A$ is a change of basis matrix whose span includes the vector v. Scaling the new basis $A$ by the values stored in $x$, followed by linearly combining the basis, will give us $v$.
Could all linear transformations also be considered as a change of basis?
Edit: A comment pointed out that a basis by definition must be a set of linearly independent vectors. So not all linear transformations are a change of basis. In that case, what's the point of a change of basis matrix? It just seems like a label for a specific type of linear transformation.
 A: In the setting of abstract (finite-dimensional, real) vector space $V$ and a linear transformation $T:V\to V$ on $V$, there is no notion of matrices involved.
If you choose an ordered basis $\beta$, then $T$ can be represented as an $n\times n$ matrix $[T]_{\beta}$.
On the other hand, if you have two bases $\alpha$ and $\beta$, given any vector $x\in V$, the two different coordinates of $x$ (wrt $\alpha$ and $\beta$),  $[x]_\alpha$ and $[x]_\beta$ are related by a matrix $A_{\alpha}^\beta$:
$$
[x]_\alpha=A_{\alpha}^\beta[x]_{\beta}\tag{1}
$$
This matrix $A_{\alpha}^\beta$ is a linear transformation on $\mathbb{R}^n$, where $n$ is the dimension of $V$. One can show that  $A_{\alpha}^\beta$ must be invertible. So the answer to your question in the title is no, since no every linear transformation (and thus its matrix under any basis) is invertible.
The change-of-basis matrix gives you a formula that expresses the coordinates relative to one basis in terms of coordinates relative to the other basis.
