• 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.