Why is a linear transformation expressed using its transpose? If $A$ is an invertible matrix with entries from $\mathbb{R}$, what is the reasoning behind defining an invertible linear transformation $f_A:\mathbb{R}^n \rightarrow \mathbb{R}^n$ as $f_A=xA^t$, where $A^t$ represents the transpose of $A$? A more "natural" definition to me would be$f_A=Ax$.

Let $f_A(x)=xA^t$ (where $x$ is a row vector) and $g_A(x)=Ax$ (where $x$ is a column vector).
Note that
$$
f_A(x)^t=(xA^t)^t=Ax^t=g_A(x^t).
$$
Hence, $f_A$ and $g_A$ are essentially the same transformation, except that $f_A$ uses row vectors and $g_A$ uses column vectors. If you begin with some $x\in\mathbb R^n$, treat it as a row vector, apply $f_A$ and then take the transpose to turn it into a column vector, you get the same result as you would if you apply $g_A$ to $x$ as a column vector.
