Linear Algebra
Column Space & Null Space
A matrix squeezes its input space onto its column space and crushes its null space to zero. Every input is a row-space part that matters plus a null-space part that vanishes, and Ax = b has a solution exactly when b lies on the column space.
c₁c₂c₃C(A), column spaceN(A), null spacerow-space partx and Axb
Drag either view to orbit · drag x, z or b · click a lattice point to pick it · scroll to zoom
Rank + nullity—
—
Matrix
Entries of A (one column per colour)
0.000.000.000.000.000.000.000.000.00
independentin the plane of c₁, c₂
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Values
x—
row-space part xᵣ—
null-space part xₙ—
Ax = Axᵣ—
b—
distance from b to C(A)—
singular values—
What to observe
- On Rank 2, c₃ = ½c₁ + ½c₂, so the green null line is every multiple of (½, ½, −1). Slide z along it: its image never leaves the origin.
- Start from Full rank and pull column 3 into the plane of c₁ and c₂. The cloud flattens onto the sheet; the moment it lands, rank drops to 2, a null line appears, and rank + nullity stays 3.
- Each x splits into a row-space part plus a null-space part. On Rank 1 the null space is a whole plane, and only the violet part decides where x lands: Ax = Axᵣ.
- In Solve mode b lights up only on the sheet. Then every solution is one particular xp plus the null line, and sliding along that line keeps Ax = b.
Shortcuts:space animate · r reset · f fullscreen