A matrix ℳ represents a sequence of + and × operations. At the end you’ve linearly transformed a space (sheared it, expanded it, rotated it — but kept the origin where it is.)

sheared Mona Lisa

Did the amount of stuff in the picture change when you did that? If you kept everything in proportion then det |ℳ| = 1. If not, then det |ℳ| ≠ 1.

image

If the amount of stuff increased by 10% then det |ℳ|=1.1. If you effectively shrank the picture in half, then det |ℳ|=.5. And so on.

The determinant |ℳ| is the change in volume after the linear transformation.

This metaphor extends to 3-D and beyond.

  • If water is flowing linearly in a stream, then |ℳ| needs to be 1, or else water (matter) would be being created.
  • If money is flowing linearly in a billion-dimensional economic system, then |ℳ| is hopefully just a little bit above 1, if value is being created. (Central banks need to print |ℳ| times more money to prevent deflation.) 
  • And a hundred-dimensional linear dynamical system's phase space grows by |ℳ| at every step.

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    I didn’t know this about determinants…. I just did the the teacher said. Math is freakin amazing.
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