Two 3×3 matrices. One squeezes R³ onto a line, the other onto a plane.
The matrix
Combine the columns
1.00
0.00
0.00
Tip-to-tail arrows. For A the path is glued to a line; for B it can wander around a whole plane — but never off it.
Feed in a vector x, watch Ax
1.0
1.0
0.0
Amber = Ax, always inside the column space. Cyan = the piece of x that A throws away (it lies in the null space).
readout
show
the real difference
A
B
rows
1 3 8 1 3 8 1 3 8
1 2 3 2 4 6 1 1 1
independent columns
1
2
rank
1
2
column space
LINE
PLANE
null space
plane
line
nullity
2
1
rank + nullity = 3 always. A's columns are all parallel, so only one direction survives and two dimensions of input are crushed. B's columns spread out into a plane, so two directions survive and only one direction of input is crushed. Rank is the number of independent directions you can actually reach.