• Discussion of subspaces plane $$P$$ through zero and line $$L$$ through zero in $$R^3$$. A bit confusing the exact definition of union and intersection, but if union is not a subspace in general (assuming $$L$$ is not coplanar with $$P$$) while intersection is.

• Interestingly: overdetermined equations (in this example, 3 unknowns and 4 equations) don’t fill the $$R^4$$ space so they can’t always be solved. Overdetermined doesn’t fill the subspace in this case.

• Column space : all $$b$$’s that solve $$Ax=b$$. In an $$m \times n$$ matrix, column space is a subspace $$R^m$$

• Nullspace: all $$x$$’s that solve $$Ax=0$$. In an $$m \times n$$ matrix, null space is a subspace of $$R^n$$

• Note that the solutions to $$Ax=b$$ do not form a subspace in general for $$b$$ (generally will not include the zero vector). The nullspace is a vector space.

• Previewing the idea that you will have a particular solution to $$Ax=b$$, but then you can add a vector from the nullspace to also get another solution.