Subspaces of Rⁿ

A subspace is simply a line, plane or higher-dimensional flat sheet that passes through the origin. The "sub" means it's a smaller space living inside a bigger one, like a flat tabletop inside the 3D room, and it still behaves like a space of its own. Subspaces are good for describing exactly what a matrix can reach and what it throws away. The two formulas look technical, but each is one idea: $\operatorname{Col}A$ (the column space, every output $A\vec x$ can produce, which is all the combinations of $A$'s columns) and $\operatorname{Nul}A$ (the null space, every input that $A$ sends to zero). Two sets, one about outputs and one about inputs. That's it.

Testing a set. Ask three questions. Does it contain $\vec 0$? Is the sum of two members a member? Is every multiple of a member a member? Think about why lines or planes that miss the origin always fail: multiply any member by $0$ and you land on $\vec 0$, which isn't there. There's an animation further down that shows a shifted line failing exactly this way.

Bases. You met spans and independence earlier. A basis of a subspace puts them together: an independent set that spans it, the smallest set of directions you need to reach everything.

• For $\operatorname{Col}A$: the pivot columns of $A$ itself.
• For $\operatorname{Nul}A$: solve $A\vec x = \vec 0$ in parametric form, and take the vectors multiplying the free variables.

A common mistake. It's tempting to take the pivot columns of the echelon form as the basis for $\operatorname{Col}A$, since that's where you found the pivots. It isn't right: row operations change the columns, so the echelon form's columns usually span a different space. Use the echelon form to find which columns have pivots, then go back and take those columns from $A$.

In signal processing, the "noise subspace" and "signal subspace" are separated to clean up data; in recommender systems, user preferences are modeled as living in a low-dimensional subspace.