Two Landings, One Distance

A linear transformation $T$ acts on the whole plane. You are never shown its formula — all you are told is where the two basis vectors land:

Input vector	Lands at
$\hat{\imath} = (1, 0)$	$(3, -1)$
$\hat{\jmath} = (0, 1)$	$(-2, 4)$

A data point sits at $v = (5, 3)$ .

Work out where $T$ sends $v$ , then report the length of that image vector:

$\lVert T(v) \rVert$

Round your answer to two decimal places.