Null Space and 3D Transformations
A 3×3 matrix moves every point of ℝ³. The null space is the set of inputs it sends to zero, and its size tells you how much the transformation flattens space.
Columns as transformed axes
The columns of a 3×3 matrix A are where it sends the three basis vectors. Every output Ax is a combination of those columns, so the image is their span: all of space if they are independent, a plane if the rank is 2, or a line if the rank is 1.
The null space
The null space contains every x with Ax = 0. Take A with rows (1, 2, 3), (2, 4, 6) and (1, 0, 1). Row reducing gives x + 2y + 3z = 0 and y + z = 0, so y = −z and x = −z. Every solution is a multiple of (−1, −1, 1), a line through the origin. Check the first row: −1 − 2 + 3 = 0.
Rank–nullity
Rank plus nullity equals the number of columns. Here the rank is 2 and the nullity is 1: the matrix crushes a whole line of inputs to the origin and squashes the rest of space onto a plane. In the 3D view, the null-space line is exactly the direction that gets collapsed.
Common mistake
The null space lives in the input space, not the output space. Do not confuse it with the column space, which is the set of outputs.
Why it matters for equations
If Ax = b has one solution x₀, then every solution is x₀ plus a null-space vector. A null space of just the zero vector means the solution is unique. A nonzero null space means infinitely many solutions, and for a square matrix it also means the determinant is zero.
FAQ
Is the null space always a line?
No. Its dimension is the nullity: a point, a line, a plane or all of ℝ³, depending on the rank.