The Null Space
The null space is the flip side of the range: it's every input that gets crushed to the origin (zero). In the shadow picture, it's the directions you can slide the object without the shadow moving at all — the directions the transformation is completely blind to.
⚠️ Don't mix these two up — they live in different places. The range is in the output space (where things can land). The null space is in the input space (where things come from). Range = where it can go; null space = what it erases on the way.
Watch the INPUT panel below. Switch to Q and drag r along the red dashed line: the output g doesn't budge at all. That whole red line of inputs collapses to the same output — and the part of it through the origin's direction collapses to zero. That red line is the null space. For P, there's no such line — only the origin maps to the origin, so P's null space is just a single point.
The red dashed line is the null space: every input on it lands on the same output. Drag r along it — the output g doesn't move! The output only ever lands on the green range line (rank 1). So given g, you can't tell which r it came from → not invertible, information lost.
The rule that ties it together: if the null space is bigger than a single point (dimension > 0), the transformation loses exactly that many dimensions. Q's null space is a 1-D line → it loses 1 dimension → 2-D squished onto 1-D. A null space of just the origin means nothing is lost.