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The determinant of the matrix of its vectors gives the measure of an $n$-dimensional parallelogram.

For example, in $2$ dimensions, the area spanned by vectors $v$ and $w$ is \begin{array}{|cc|} v_1 & w_1 \\ v_2 & w_2 \\ \end{array} and so forth for a $3$ or more -dimensional parallelogram.

How is possible to visualize that, or understand that intuitively?

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Have a look at "Linear Algebra via Exterior Products" by S. Winitzki: sites.google.com/site/winitzki/linalg , pagg.30-31. –  Giuseppe Negro Apr 30 '12 at 16:53
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1 Answer 1

You can see the determinant when changing variables of integration in two or three dimensions. The determinant can be viewed as the Jacobian of the transformation

$$ (x,y) \mapsto (v_1x+v_2y,w_1x+w_2y)$$

and this maps the unit square onto the parallelogram spanned by $(v_1,v_2)$ and $(w_1,w_2)$. The unit square has area 1 and therefore the determinant balances the equation.

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