• 4 fundamental subspaces (assume \(A\) is \(m \times n\)):
    • Columnspace: \(C(A)\) is in \(R^m\)
    • Nullspace: \(N(A)\) is in \(R^n\)
    • Rowspace: all combinations the rows of \(A\) (i.e. \(C(A^T)\)) is in \(R^n\)
    • Left Nullspace: \(N(A^T)\) is in \(R^m\)
  • Dimension of the subspaces:
    • Columnspace: \(rank(A)\)
    • Nullspace: \(n - rank(A)\)
    • Rowspace: \(rank(A)\)
    • Left Nullspace: \(m - rank(A)\)

  • Note the sum of the dimensions of the nullspace and rowspace give \(n\) (and they are both in \(R^n\)) and the sum of the dimensions of the columnspace and left nullspace give \(m\) (and they are both in \(R^m\)).

  • How to produce a basis for each subspace:
    • Columnspace: row reduction, use the original vectors that correspond to the pivot columns.
    • Nullspace: set each free variable to 1 (and others to zero) to find basis vectors (i.e. find the special solutions).
    • Rowspace: the pivot rows directly after getting \(A\) into rref.
    • Left Nullspace: row reduce \(A^T\) and find the special solutions.