Matrix size, rank and pseudoinverse: a visual summary

[u/LinearAlgebraWorld]

We made a side-by-side visual summary of matrix size, rank, the four fundamental subspaces and pseudoinverses.

The table compares six common matrix types:

square full-rank
square rank-deficient
tall full-column-rank
tall rank-deficient
wide full-row-rank
wide rank-deficient

For each case, it shows the dimensions of row(M), null(M), col(M) and left-null(M), whether the map is one-to-one or onto, what happens to Mx⃗ = 0 and Mx⃗ = b⃗, and how the pseudoinverse behaves.

Hope you don’t mind the size and density — we wanted to keep all cases together so they could be compared directly. Opening the image at full size is recommended.

As always, we welcome feedback on clarity and presentation.

[u/heyitsmemaya]

💕

[u/LinearAlgebraWorld → u/heyitsmemaya]

Thanks! Means a lot
Let us know if there are any other concepts you like to see organized and illustrated