• 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.