Meaning of $L_A$?

Let $A$ be an m*n matrix with entries from a field $F$.
$L_A: F^n \rightarrow F^m$ defined by $L_A=Ax$.

$L_A$ is a matrix representation of linear transformation $T: V \rightarrow W$ where $V, W$ are vector spaces with dimension $n, m$, respectively?

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$A$ is the matrix. $L_A$ is the function $L_A:F^n\rightarrow F^m$ defined by $L_A(x)=Ax$.
In other words, $A$ is the actual matrix itself, and $L_A$ is the function which applies the matrix. (As it turns out this function is linear, i.e. $L_A(ax+by)=aL(x)+bL(y)$, which is why they call it a linear transformation.)
Think of it like this: $A$ is the hammer, whereas $L_A$ is the act of swinging it.