The rank-nullity theorem

Kernel and image do not vary independently: what is lost on one side is exactly what is missing on the other. This is the rank-nullity theorem, the most used result of the whole course.

The statement

For f linear from a finite-dimensional space E to a space F:

dim Ker f  +  dim Im f  =  dim E
              \________/
                rank of f

The dimension of the image is called the rank of f. It is the same rank as that of the matrix of f — the number of pivots.

The intuition: a dimension budget

The source space has dim E dimensions. Each is either crushed (it goes into the kernel) or transmitted (it contributes to the image). Nothing is lost, nothing is created:

   dim E = 3
   ______________________________
  |          |                   |
  | crushed  |    transmitted    |
  | (kernel) |     (image)       |
  |__________|___________________|
      dim 1          dim 2

The projection example

The projection of R³ onto the plane z = 0 has matrix diag(1, 1, 0):

        z
        |    • v = (x ; y ; z)
        |   /|
        |  / |          the z-axis is crushed    -> Ker, dimension 1
        | /  |          the plane z = 0 is reached -> Im, dimension 2
    ----+----•-------
       /     p(v) = (x ; y ; 0)

               1 + 2 = 3 = dim R³      ✔

The most useful special case

When source and target have the same finite dimension — in particular for a square matrix — the theorem yields a remarkable equivalence:

f injective   <=>   f surjective   <=>   f bijective

Indeed, if Ker f = {0} then dim Im f = dim E = dim F, so Im f = F. A single check suffices, usually the kernel one, which is quicker.

Careful: this equivalence is false in infinite dimension. On the space of polynomials, differentiation is surjective but not injective (constants are crushed).

What the rank reveals immediately

For a matrix A of size m × n (so f : R^n -> R^m) of rank r:

dim Ker = n - r      number of free parameters of the system A X = 0
dim Im  = r          number of independent columns

r = n   ->  injective        (no free parameter)
r = m   ->  surjective       (the image fills the target)
r = n = m -> bijective, A is invertible

We recover exactly the results of the course on systems: n - r was already the number of parameters of the solutions. That was no coincidence — it was the rank-nullity theorem, before it had a name.

Summary

• Rank-nullity theorem: dim Ker f + rank f = dim E.
• The rank is the dimension of the image, equal to the number of pivots of the matrix.
• Every dimension of the source is either crushed or transmitted.
• In equal dimensions (square matrix): injective ⟺ surjective ⟺ bijective.
• That equivalence fails in infinite dimension.
• For A of size m × n and rank r: dim Ker = n - r, dim Im = r.