Strang §3.5 · Dimensions of the four subspaces
Four subspaces, one big picture.
Every matrix has four subspaces. The row space and the nullspace live among the inputs. The column space and the left nullspace live among the outputs. The rank r fixes all four sizes.
inputs (ℝⁿ): row space r + nullspace n − r | outputs (ℝᵐ): column space r + left nullspace m − r
Think: a shadow
The sun is straight overhead. Every object casts a shadow on the floor. That is a matrix taking 3D inputs to 2D outputs.
- Moving an object straight up or down does not change its shadow. That lost direction is the nullspace.
- Moving it sideways along the floor moves the shadow one-for-one. That part of the input is the row space.
- Every place a shadow can fall is the column space.
- If the outputs had a direction no shadow could ever reach, that would be the left nullspace, N(Aᵀ).
The dimensions
For an m by n matrix of rank r:
row space C(Aᵀ): r nullspace N(A): n − r together: n
column space C(A): r left nullspace N(Aᵀ): m − r together: m
Rows and columns always have the same rank. Strang calls these four dimensions Part 1 of the Fundamental Theorem of Linear Algebra.
Why the row space is perpendicular to the nullspace
Ax = 0 says: each row of A, dotted with x, gives 0. So x is perpendicular to every row, and to every combination of rows. Chapter 4 builds on this.
Worked example
A = 1236 has rank 1.
row space: line through (1, 2) nullspace: line through (−2, 1)
column space: line through (1, 3) left nullspace: line through (−3, 1)
Each pair is perpendicular: (1, 2)·(−2, 1) = 0 and (1, 3)·(−3, 1) = 0 ✓
The 2 by 4 matrix from slide 3 has r = 2: row space 2, nullspace 2 (the two special solutions), column space 2 (all of ℝ²), left nullspace 0.