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

I'm a bit confused about this definition.
$L_A$ 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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up vote 1 down vote accepted

$A$ 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.

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+1 for the last sentence! –  EuYu Apr 5 '13 at 5:24
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