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 14 of 22

Running it backwards

Understand first: determinant · rank and collapse

Every system of linear equations is secretly the same question. Take

2x+y=5,x+3y=52x + y = 5, \qquad x + 3y = 5

and write it as a single transformation applied to an unknown vector:

[2113][xy]=[55]\begin{bmatrix} 2 & 1 \ 1 & 3 \end{bmatrix}\begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 5 \ 5 \end{bmatrix}

Now read it aloud in the language you have built: which starting vector, after this motion, lands on (5,5)(5,5)?

When you can just play it in reverse

If the motion did not destroy anything, there is another linear transformation that undoes it, called the inverse and written A−1A^{-1}. Run AA then A−1A^{-1} and every vector returns home, so together they do nothing at all:

A−1A=I=[1001]A^{-1}A = I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}

The identity matrix II is the do-nothing motion: its columns say ı^\hat{\imath} lands on ı^\hat{\imath}, and ȷ^\hat{\jmath} on ȷ^\hat{\jmath}.

With an inverse in hand, the system is solved by playing the motion backwards from the target: x=A−1vx = A^{-1}v. Our example has det⁡=2(3)−1(1)=5≠0\det = 2(3) - 1(1) = 5 \neq 0, so it works and gives (x,y)=(2,1)(x, y) = (2, 1). Check it: 2(2)+1=52(2) + 1 = 5 and 2+3(1)=52 + 3(1) = 5.

When it is impossible

Watch the visual collapse the plane onto a line. Many different starting vectors now land on the same point.

Ask the reverse question at one of those points and it has no single answer — several inputs are equally valid. A function cannot return several answers, so no inverse transformation exists.

This is exactly the case det⁡A=0\det A = 0, exactly the case where rank dropped. Three descriptions, one phenomenon.

And the collapse leaves a specific fingerprint: a whole set of vectors got crushed onto the origin. Naming that set is the last piece of Act III.