MAT 102 Linear Algebra · DSE · Intake 2/1

The Four Fundamental Subspaces

Factor any matrix as \(A = CR\) with Gauss–Jordan elimination: reduce to \(R_0 = \mathrm{rref}(A)\), keep the pivot columns of \(A\) in \(C\), the nonzero rows of \(R_0\) in \(R\). Then use it everywhere — the column space and when \(A\mathbf{x}=\mathbf{b}\) is solvable, the nullspace and its special solutions, the complete solution \(\mathbf{x}_p + \mathbf{x}_n\), independence and bases, and the dimensions \(r,\, r,\, n-r,\, m-r\) of all four subspaces.