Strang §3.2 · The nullspace of A
The nullspace: inputs that do nothing.
The nullspace N(A) holds every x with Ax = 0. Feed in any of these inputs and nothing comes out.
N(A) = all x with Ax = 0
Think: engines running, boat standing still
Back to the boat. Thruster 2 points the same way as thruster 1 and pushes twice as hard. So A = 1224.
Set thruster 1 forward at 2 and thruster 2 in reverse at 1. The pushes cancel exactly. Both engines are working, and the boat does not move. The throttle setting (2, −1) is in the nullspace.
What is always true
- x = 0 is always in N(A).
- N(A) is a subspace: if Ax = 0 and Ay = 0, then A(cx + dy) = 0 too.
- If A is invertible, N(A) is only the zero vector.
- A nonzero x in N(A) is a recipe for a dependency: x₁·(column 1) + x₂·(column 2) = 0.
Worked example
A(2, −1) = (1·2 + 2·(−1), 2·2 + 4·(−1)) = (0, 0) ✓
Every multiple c·(2, −1) works too, so N(A) is a whole line. Look at the plot: it is perpendicular to the row (1, 2). That is no accident, and slide 6 explains it.
Where you meet this
A robot arm with more joints than it needs has a nullspace. Joint motions in that nullspace move the elbow while the hand stays exactly where it is. Engineers use it to steer the elbow around obstacles while the tool keeps working.