Fundamental Theorem of Linear Algebra

• categories: Linear algebra, Theorem

4 Fundamental Subspaces of $A$ ($m\times n$) with the Rank$=r$

1. Column Space $C(A)$ $dim=r$
2. Row Space $C(A^T)$ $dim=r$
3. null space $N(A)$ $dim=n-r$
4. null space $N(A^T)$ $dim=m-r$

Theorem :

Column Rank equals row Rank

Observation :

Column/row spaces are orthogonal to the corresponding null spaces

Visualization :