intuition.
  1. 1
  2. 2
  3. 3
  4. 4
  5. 5
  6. 6
  7. 7
  8. 8
  9. 9
  10. 10
  11. 11
  12. 12
  13. 13
  14. 14
  15. 15
  16. 16
  17. 17
  18. 18
  19. 19
  20. 20
  21. 21
  22. 22
Linear Algebra
Step 12 of 22

Determinant as volume

Understand first: determinant · transformations 3d

In the plane the determinant scaled area. Add a dimension and the story is identical with one word swapped.

Drag to orbit. The unit cube sits on ı^\hat{\imath}, ȷ^\hat{\jmath}, k^\hat{k} with volume 1. The transformation turns it into a slanted box, and the determinant is the factor its volume changed by.

A case you can verify

M=[200030001]M = \begin{bmatrix} 2 & 0 & 0 \ 0 & 3 & 0 \ 0 & 0 & 1 \end{bmatrix}

Read the columns: ı^\hat{\imath} doubles, ȷ^\hat{\jmath} triples, k^\hat{k} is untouched. The unit cube becomes a 2×3×12 \times 3 \times 1 box, volume 6. The determinant is 6, and here it is simply the product of the diagonal entries, because nothing was tilted.

Watching a dimension die

Let the visual run until the box goes flat. The readout slides toward zero and the solid becomes a sheet with no thickness at all.

That is a 3D transformation whose output is a plane. Every point in space now lands somewhere on that plane. One full dimension has been erased, and no transformation applied afterwards can restore it — the information about where things came from is simply not there any more.

Flatten it once more and you would get a line: two dimensions gone. The determinant reports zero in both cases, which is a real limitation. It tells you that a collapse happened but not how far it went, and that gap is exactly what rank exists to fill.

Sign in 3D

A negative determinant still means orientation reversed. In 3D that is the difference between a right hand and a left hand: point your fingers along the first two basis vectors, and the third either follows your right thumb or your left. A negative determinant means the transformation swapped which hand space obeys, and no amount of rotating can fix that.

Zero keeps demanding attention. Next: exactly how many dimensions survived.